The Chain Rule: Learn It 2

Combining the Chain Rule With Other Rules

The Chain and Power Rules Combined

We can now apply the chain rule to composite functions, but note that we often need to use it with other rules. For example, to find derivatives of functions of the form [latex]h(x)=(g(x))^n[/latex], we need to use the chain rule combined with the power rule. To do so, we can think of [latex]h(x)=(g(x))^n[/latex] as [latex]f(g(x))[/latex] where [latex]f(x)=x^n[/latex]. Then [latex]f^{\prime}(x)=nx^{n-1}[/latex]. Thus, [latex]f^{\prime}(g(x))=n(g(x))^{n-1}[/latex]. This leads us to the derivative of a power function using the chain rule,

[latex]h^{\prime}(x)=n(g(x))^{n-1}g^{\prime}(x)[/latex]

power rule for composition of functions

For all values of [latex]x[/latex] for which the derivative is defined, if

[latex]h(x)=(g(x))^n[/latex]

 

Then

[latex]h^{\prime}(x)=n(g(x))^{n-1}g^{\prime}(x)[/latex]

Find the derivative of [latex]h(x)=\dfrac{1}{(3x^2+1)^2}[/latex]

Find the derivative of [latex]h(x)=\sin^3 x[/latex]

Find the equation of a line tangent to the graph of [latex]h(x)=\dfrac{1}{(3x-5)^2}[/latex] at [latex]x=2[/latex].


The Chain and Trigonometric Functions Combined

Now that we can combine the chain rule and the power rule, we examine how to combine the chain rule with the other rules we have learned. In particular, we can use it with the formulas for the derivatives of trigonometric functions or with the product rule.

Find the derivative of [latex]h(x)= \cos (g(x))[/latex].

In the following example we apply the rule that we have just derived.

Find the derivative of [latex]h(x)= \cos (5x^2)[/latex].

Find the derivative of [latex]h(x)= \sec (4x^5+2x)[/latex].

At this stage, we present a collection of derivative formulas derived by applying the chain rule along with the standard derivatives of trigonometric functions. The derivation methods for these formulas are analogous to those demonstrated in the previous examples.

For ease of learning and recall, we have included these formulas in Leibniz’s notation, which some students may find more intuitive. Later in this section, we explore the application of the chain rule in Leibniz’s notation in greater detail.

It is important to note that memorizing these formulas as distinct entities is not essential; they are all manifestations of the chain rule applied to well-established derivative formulas.

using the chain rule with trigonometric functions

For all values of [latex]x[/latex] for which the derivative is defined,

[latex]\begin{array}{llll}\frac{d}{dx}(\sin (g(x)))= \cos (g(x))g^{\prime}(x) & & & \frac{d}{dx} \sin u= \cos u\frac{du}{dx} \\ \frac{d}{dx}(\cos (g(x)))=−\sin (g(x))g^{\prime}(x) & & & \frac{d}{dx} \cos u=−\sin u\frac{du}{dx} \\ \frac{d}{dx}(\tan (g(x)))= \sec^2 (g(x))g^{\prime}(x) & & & \frac{d}{dx} \tan u=\sec^2 u\frac{du}{dx} \\ \frac{d}{dx}(\cot (g(x)))=−\csc^2 (g(x))g^{\prime}(x) & & & \frac{d}{dx} \cot u=−\csc^2 u\frac{du}{dx} \\ \frac{d}{dx}(\sec (g(x)))= \sec (g(x)) \tan (g(x))g^{\prime}(x) & & & \frac{d}{dx} \sec u= \sec u \tan u\frac{du}{dx} \\ \frac{d}{dx}(\csc (g(x)))=−\csc (g(x)) \cot (g(x))g^{\prime}(x) & & & \frac{d}{dx} \csc u=−\csc u \cot u\frac{du}{dx} \end{array}[/latex]

The Chain and Product Rules Combined

When tackling calculus problems involving products of composite functions, combining the chain and product rules proves indispensable. This approach allows for the systematic differentiation of functions where both rules are necessary to compute the derivative accurately.

Find the derivative of [latex]h(x)=(2x+1)^5(3x-2)^7[/latex]