Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ \frac{d}{d x} F(\cos x) $$

Step-by-Step Solution

Answer

\(-F'(\cos x) \sin x\)

Step 1: Identify Outer and Inner Functions

We want to find \( \frac{d}{dx} F(\cos x) \). Here, \( F(u) \) is the outer function where \( u = \cos x \) is the inner function.

Step 2: Apply the Chain Rule

The chain rule states that if you have a composite function \( F(g(x)) \), then the derivative \( \frac{d}{dx} F(g(x)) = F'(g(x)) \cdot g'(x) \). Here \( g(x) = \cos x \).

Step 3: Differentiate the Outer Function

Find the derivative of the outer function with respect to the inner function: \( F'(\cos x) \). This is left as an expression since \( F \) is a general differentiable function.

Step 4: Differentiate the Inner Function

Next, find the derivative of the inner function \( \cos x \) with respect to \( x \). The derivative \( \frac{d}{dx}(\cos x) = -\sin x \).

Step 5: Combine Results Using the Chain Rule

According to the chain rule, multiply the derivatives from Steps 3 and 4: \[\frac{d}{dx} F(\cos x) = F'(\cos x) \cdot (-\sin x) = -F'(\cos x) \sin x\]

Key Concepts

Composite Function

A composite function is like putting one function inside another. Imagine you have two functions, say function \( F \) and function \( g \).
When you form a new function by using the output of \( g(x) \) as the input for \( F \), you have a composite function. It is denoted as \( F(g(x)) \).
This is similar to a matryoshka doll where one doll fits inside another.

• The inner function in our exercise is \( g(x) = \cos x \).
• The outer function is \( F(u) \) where \( u = \cos x \).

Recognizing these inner and outer functions helps us differentiate composite functions efficiently.

Differentiation

Differentiation is the process we use to find the derivative of a function. It's like finding the slope of the function at any point along its curve.
In the context of the exercise, we want the slope of the composite function \( F(\cos x) \).
To do this, the chain rule becomes very handy. It tells us how to differentiate a composite function by relating the derivatives of its inner and outer functions.
First, you differentiate the outer function keeping the inner function unchanged. Then you multiply it by the derivative of the inner function. Differentiation helps us understand how the change in one quantity leads to change in another.

Derivative

The derivative gives us the rate at which one quantity changes with respect to another.
In simpler terms, it's the speed of change at any point for a function on its graph.
For the function \( F(\cos x) \), finding the derivative involves two steps:

• First, find the derivative of the outer function \( F(u) \). This is \( F'\) evaluated at \( u = \cos x \).
• Next, find the derivative of the inner function \( \cos x \), which is \(-\sin x\).

Putting these together using the chain rule gives us the final derivative: \(-F'(\cos x) \sin x\).
This result effectively gives the rate of change of \( F(\cos x) \) with respect to \( x \).