Chapter 3 The Derivative

Author

Jiaye Xu

Published

October 6, 2025

3.1 Two Problems with One Theme

The two problems are identical twins:

  • the problem of the slope of the tangent line;

  • the problem of instantaneous velocity.

The Tangent Line

The tangent line is the limiting position of the secant line. The secant line through \(P\) and \(Q\) has slope \(m_{sec}\) given by

Example 1, 2, 3 , page 120. Find the slope of the tangent line by definition, using the limit finding techniques from chapter 2.

Average Velocity and Instantaneous Velocity

Average velocity is the distance from the first position to the second position divided by the elapsed time. At time \(c\) the object is at \(f(c)\); at the nearby time \(c+h\), it is at \(f(c+h)\).Thus the average velocity on this interval is

Comments: Now you see why we called this section “Two Problems with One Theme.” Look at the definitions of slope of the tangent line and instantaneous velocity.They give different names for the same mathematical concept.

Example 4 & 5, page 122. In the case where \(f(t)=16t^2\), find the instantaneous velocity by definition. And solve for time \(c\).

Example 6, page 122. A little trick for finding the limit – multiply top and bottom by the conjugate of the numerator, i.e., \(\sqrt{16+5h}+4\).

Rates of Change

Velocity is only one of many rates of change that will be important in this course; it is the rate of change of distance with respect to time. The phrase rate of change without an adjective will mean instantaneous rate of change.

3.2 The Derivative

Besides the slope of the tangent line and instantaneous velocity, other versions of the same concept include:

  • rate of growth of an organism in biology;

  • marginal profit in economics;

  • density of a wire in physics;

  • dissolution rates in chemistry.

Good mathematical sense suggests that we study this concept independently of these specialized vocabularies and diverse applications.We choose the neutral name derivative. Add it to function and limit as one of the key words in calculus.

Example 1-3, page 126. Find the derivatives via definition formula.

Example 4, page 127. Now you must be quite familiar with the multiplying-the-conjugate technique for rationalizing the numerator and finding the limit.

Equivalent Forms for the Derivative

N.B. \(c\) is a particular \(x\), which is fixed when you evaluate the derivative. \(h\) is a dummy variable. You are free to use different letters.

Example 5, page 128. Cancel the common factor, evaluate by substitution.

Example 6, page 128. (a) By the original version of definition, \(f(x)=x^2=16\). at \(x=4\). (b) By the equivalent version of definition, \(f(x) =2/x\), at \(x=3\).

Differentiability Implies Continuity

If a curve has a tangent line at a point, then that curve cannot take a jump or wiggle too badly at the point. The precise formulation of this fact is an important theorem.

Go over the proof in class. The fancy way of writing \(f(x)\), \[f(x)=f(c)+\frac{f(x)-f(c)}{x-c}\cdot(x-c), \quad x\neq c\]

Bear in mind this form, you will see this equation in similar form multiple times down the road of learning, like in linearization and Taylor Series.

N.B. The converse of this theorem is false. Example of \(f(x)=|x|\) at the origin. The derivative in the form of limit does not exist, which is shown via different right- and left-hand limits.

At any point where the graph of a continuous function has a sharp corner or where the tangent line is vertical, the function is not differentiable.

Increments and Leibniz Notation for the Derivative

If the value of a variable \(x\) changes from \(x_1\) to \(x_2\), then \(x_2-x_1\), the change in \(x\), is called an increment of \(x\) and is commonly denoted by \(\Delta x\).

Comments: \(\Delta x\) is just another notation for \(h\). Bear with the notations, where the beauty of math language lies.

Example 7, page 129. \(\Delta y\) is \(y_2-y_1\), i.e., \(f(x_2)-f(x_1)\).

Now we introduce the Leibniz notation, a standard symbol for the derivative, and we will use it frequently from now on.

The Graph of the Derivative

When the tangent line is sloping up to the right, the derivative is positive, and when the tangent line is sloping down to the right, the derivative is negative. We can therefore get a rough picture of the derivative given just the graph of the function.

3.3 Rules for Finding Derivatives

The process of finding the derivative of a function directly from the definition of the derivative can be time consuming and tedious. We are going to develop tools that will allow us to shortcut this lengthy process.

\(D_x\) is an example of an operator. The derivative operates on \(f\) to produce \(f^\prime\). We now have three notations for the derivative.

The Constant and Power Rules

The constant function is a horizontal line, which therefore has slope zero everywhere.

proof by definition.

The graph of the identity function is a line through the origin with slope 1; so we should expect the derivative of this function to be 1 for all x.

Go over the proof. Recall the binomial expansion formula.

\(D_x\) Is a Linear Operator

Example 1, page 136.

Product and Quotient Rules

Example 2, page 136. Give us an example of a common mistake when applying product rule.

Example 3, page 137.

Memorize in words: The derivative of a quotient is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Example 4, page 138.

Example 5, page 139.

3.4 Derivatives of Trigonometric Functions

Recall the Addition Identity for \(\sin(x+h)\), and show \(D_x(\sin x)\) via definition formula. Key limits result in Section 2.5 are used in this proof.

Example 1, page 141. Apply Thm A.

Example 2, page 141. To find the tangent line, we need the slope, and one point that the line goes through. Point-Slope formula.

curve(3*sin(x), 0, 2*pi)
abline(h=0, col="grey")
abline(b=-3, a=3*pi)

The Product and Quotient Rules are useful when evaluating derivatives of functions involving the trigonometric functions.

Example 3 & 4, page 142.

Example 5, page 142. A reading comprehension trick: if you don’t quite understand what “a bobbing cork” is, just regard it as some object what-you-may-call-it. The key idea to solve this problem is what you are very familiar with: The velocity is the derivative of position.

Since the tangent, cotangent, secant, and cosecant functions are defined in terms of the sine and cosine functions, the derivatives of these functions can be obtained from Theorem A by applying the Quotient Rule.

Example 6 & 7, page 142.

Example 8, page 143. Find the derivative and set it to 0. The solution is when either sin x or cos x is equal to zero; that is, at \(k\pi\), \(\frac{\pi}{2}+k\pi\), for \(k\in \mathbb Z\).

3.5 The Chain Rule

In last example, we have a way to make the calculation simpler – The Chain Rule is so important that we will seldom again differentiate any function without using it.

Comments: I cannot overemphasize the Chain Rule. You will benefit from this theorem not just in this semester, also for next semester in multivariable calculus, and even long-term-wise learning, e.g., back-propagation algorithm in deep learning.

Differentiating a Composite Function

You can remember the Chain Rule this way: The derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function.

Applications of the Chain Rule

Go over example 1-7, page 145 to 147.

As a general rule, if the numerator of a fraction is a constant, then do not use the Quotient Rule; instead write the quotient as the product of the constant and the expression in the denominator raised to a negative power, and then use the Chain Rule.

Example 8, page 147. Tell apart the outer function and the inner function. Recall the rule that the last step in calculation corresponds to the first step in differentiation.

Applying the Chain Rule More than Once

Go over example 9-11, page 148.

Example 11, approximation via graphs (toy example). Be careful with the composite function \(f\circ g\), \(f(\cdot)\) is the outer function, \(g\) is the inner function.

Proof of the Chain Rule (Optional)

A partial proof in textbook.

Comment: A rigorous proof of the chain rule is provided in A.2, Theorem B, in the Appendix. You may like to read it when you finish reading section 12.6.

3.6 Higher-Order Derivatives

Displacement, Velocity and Acceleration

Example 2, page 152, tells us that velocity has a sign associated with it; it may be positive or negative.

N.B. The distance is the absolute value of the displacement, with the displacement defined as \[\text{displacement}=\text{final position}-\text{initial position}.\] Speed is the absolute value of the velocity. The rate of change of velocity with respect to time is acceleration. If it is denoted by \(a\), then \[a=\frac{dv}{dt}=\frac{d^2s}{dt^2}\] In Example 2, \(s = 2t^2 - 12t + 8\), thus, \(v=4t-12, a=4\).This means that the velocity is increasing at a constant rate of \(4\) centimeters per second every second, i.e., \(4\text{ cm/sec}^2\).

Example 3, page 153.

Falling-Body Problems

You may feel confused about the the formula for the position of a falling object given in the textbook, \[s=-16t^2+v_0t+s_0,\] which is correct when the gravitational constant \(g\) is in the unit of feet/sec\(^2\). Note that \(9.8\) m/s\(^2\) \(\approx 32\) ft/s\(^2\), and the formula for the position of this falling object is \[s=-\frac{1}{2}gt^2+v_0t+s_0.\]

N.B. Metric system is used officially in science and almost all countries.

Example 4, part (a), the ball reached its maximum height at the time its velocity was 0. part (c), the ball hit the ground when \(s=0.\) (e) the acceleration is a constant.

3.7 Implicit Differentiation

When the equation defines \(y\) as an implicit function of \(x\), how do we find the derivative w.r.t \(x\), i.e., \(\frac{dy}{dx}\)?

  • Method 1 Solve the given equation explicitly for \(y\).

  • Method 2 Implicit Differentiation. Take derivative of both sides and equate the results.

Example 1, page 157.

Note that it seems that these two methods give us different answers, but if you substitute y into the expression to find the expression in terms of x only – the two answers are identical.

Comments: While the subject of implicit functions leads to difficult technical questions (treated in advanced calculus), the problems we study have straightforward solutions.

More Examples

Example 2 & 3, page 158. Use the method of implicit differentiation.

The Power Rule Again

We have learned that \(D_x(x^n)=nx^{n-1}\) where \(n\) is any nonzero integer.We now extend this to the case where \(n\) is any nonzero rational number.

N.B. If \(x\leq 0\), then the formula holds whenever the derivative, \(x^r\), and \(x^{r-1}\) all exist.

Skip the proof in textbook.

The General Version of Power Rule is

If \(r\) is any real number, then \[\frac{d}{dx}x^r=r x^{r-1},\] for all \(x\) where the powers \(x^r\) and \(x^{r-1}\) are defined.

Proof

Differentiating \(x^r\) with respect to \(x\) gives \[\frac{d}{dx}x^r=\frac{d}{dx}e^{r\ln x}=e^{r\ln x}\frac{d}{dx}(r\ln x)=x^r\cdot \frac{r}{x}=r x^{r-1}.\] Therefore, whenever \(x>0\), \[\frac{d}{dx}x^r=r x^{r-1}.\] For \(x<0\), if \(y = x^r\), \(y^\prime\), and \(x^{r-1}\) all exist, then \[\ln |y|=\ln |x|^r\] Using implicit differentiation (which assumes the existence of the derivative \(y^\prime\)) with the fact the derivative of \(\ln |y|=\frac{1}{y}, y\neq 0\), we have \[\frac{y^\prime}{y}=\frac{r}{x}.\] Solving for the derivative, \[y^\prime=r\frac{y}{x}=r\frac{x^r}{x}=rx^{r-1}.\] The derivative equals \(0\) when \(x = 0\). This completes the proof of the general version of the Power Rule for all values of x.

Q.E.D.

Example 4, page 159.

3.9 Derivatives of Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverses of one another, so to get their derivatives, as well as the derivatives of the inverse trigonometric functions, we begin by studying inverse functions more thoroughly.

Inverse Functions

We will make use of this theorem as we use calculus to help us graph functions. Along with the result stated above that a monotonic function has an inverse, we use this theorem to determine whether a given function has an inverse, e.g., Example 1, page 169, by taking the derivative.

The conclusion to Theorem B is often written symbolically as \[\frac{dx}{dy}=\frac{1}{dy/dx}\]

Example 3, page 170. Find the derivative of the inverse in two ways – find the inverse and take derivative brute-forcely, and apply the Inverse Function Theorem.

Derivatives of Exponential and Logarithmic Functions

Armed with Theorems A and B, we are now ready to find formulas for the derivatives of the logarithmic and exponential functions. These results are given in Theorem C.

Proof.

Example 4, page 171.

In general, we have the derivative formulas for the exponential function: \[D_xa^x=a^x\ln a, \quad D_x\log_a x=\frac{1}{x\ln a}\]

Example 5, page 171.

Logarithmic Differentiation

The labor of differentiating expressions involving quotients, products, or powers can often be substantially reduced by first applying the natural logarithm function and using its properties. This method, called logarithmic differentiation.

The Functions \(a^x\), \(x^a\), and \(x^x\)

The exponential function, power function, and a variable-to-a-variable-power function.

The rigorous proof of the Power Rule is in Section 3.7.

Example 7, page 162. Find the derivative of \(x^x\) by two different methods.

Example 8, page 163. Apply the power rule, general exponential derivative and the chain rule.

Example 9, page 163. We see a power in the function, we may use logarithmic differentiation to reduce the work.

3.10 Derivatives of Inverse Trigonometric Functions

A quick review of the hyperbolic functions.

Derivatives of Hyperbolic Functions (Optional)

The results are summarized in Theorem A.

Example 1 & 2, page 175. A practice of Thm A.

A review of the inverse hyperbolic functions from section 2.6:

Each of these functions is differentiable:

Example 3, page 176. Method 1, implicit differentiation. You may need the identity for the hyperbolic functions in section 2.6, \(\cosh^2x-\sinh^2x=1\). Method 2, start with the definition expression of \(\sinh^{-1}x\).

An application of the hyperbolic function: Catenary.

If a homogeneous flexible cable or chain is suspended between two fixed points at the same height, it forms a curve called a catenary. In a coordinate system its equation takes the form \[y=a\cosh \frac{x}{a}\]

Inverse Trigonometric Functions

Proof (Skipped). Same pattern. We apply the implicit differentiation in each case. Be careful with \(D_x\sec^{-1}x\), the slope of \(\sec^{-1}x\) is always positive, while the domain could be negative, so we need an absolute sign in the result.

Example 4 & 5, page 177.

Example 6, page 178. Graph and equations on board. Use the result of \(D_x\tan^{-1}x=\frac{1}{1+x^2}\)

Summary

In this and previous sections, we have seen a number of derivative formulas. For reference, some of these formulas are listed in the table below.

3.11 Differentials and Approximations

Up to now, we have treated \(dy/dx\) (or \(d/dx\)) as a single symbol and have not tried to give separate meanings to the symbols \(dy\) and \(dx\). In this section we will give meanings to \(dy\) and to \(dx\).

Differentials Defined

Figure 2 indicates, the quantity \(dy\) is equal to the change in the tangent line to the curve at \(P\) as \(x\) changes from \(x_0\) to \(x_0+\Delta x\). When is \(\Delta x\) small, we expect \(dy\) to be a good approximation to \(\Delta y\).

Example 1, page 180. Find \(dy\) through calculating the derivative and multiply it by \(dx\).

Comments:

  1. We can interpret the derivative as a quotient of two differentials.

  2. Corresponding to every derivative rule, there is a differential rule obtained from the former by “multiplying” through by \(dx\).

Approximations

\(f(x+\Delta x)\) is approximated by

\[f(x+\Delta x)\approx f(x)+dy=f(x)+f^\prime(x)\Delta x\]

as shown in Figure 3.

Example 2, page 181. Approximate \(\sqrt{4.6}\) without a calculator. That is, approximate \(y=\sqrt x\) at \(x=4\) with \(dx=0.6\).

Example 3, page 182. Use differentials to approximate the increase in the area of a soap bubble when its radius increases from 3 inches to 3.025 inches.

Recall the area of a spherical surface is \(A=4\pi r^2\). We may approximate the exact change \(\Delta A\), by the differential \(dA\),

Estimating Errors

In scientific research, differentials are used to estimate the error. Here are two examples:

Example 4, page 182. Estimate the absolute error and the relative error of volume of the cube. The side of a cube is measured as 11.4 centimeters with a possible error of \(\pm 0.05\) centimeter.

Example 5, page 182. Given the relative change in V, find the relative change in R. Key equation by Poiseuille’s Law for blood flow, which says that the volume flowing through an artery is proportional to the fourth power of the radius, that is, \(V=kR^4\).

By how much must the radius be increased in order to increase the blood flow by 50%?

Linear Approximation

\[L(x) = f(a) + f^\prime(a)(x - a)\] is called the linear approximation to the function \(f\) at \(a\), and it is often a very good approximation to f when \(x\) is close to \(a\).

Comment: We have learnt the approximation by differential, it is natural to write out the linear approximation. \((a, f(a))\) is a particular point, \(f\) is differentiable at \(a\), note that \(\Delta x\) is \(x-a\).

Example 6, page 183.