Problem 233 - Calculus Volume 1

Question

For the following exercises, find \(\frac{d y}{d x}\) for each function. $$y=(\tan x+\sin x)^{-3}$$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = -3(\tan x + \sin x)^{-4}(\sec^2 x + \cos x)\)

Step 1: Identify the function type

The given function is a composition of two functions: an inner function and an outer function. The inner function is \(u = \tan x + \sin x\) and the outer function is \(y = u^{-3}\)

Step 2: Find the derivative of the inner function

Differentiate the inner function \(u = \tan x + \sin x\). The derivative of \(\tan x\) is \(\sec^2 x\), and the derivative of \(\sin x\) is \(\cos x\). Thus, \(\frac{du}{dx} = \sec^2 x + \cos x\).

Step 3: Find the derivative of the outer function

The outer function is \(y = u^{-3}\). Using the power rule, the derivative is \(\frac{dy}{du} = -3u^{-4}\).

Step 4: Apply the chain rule

To find \(\frac{dy}{dx}\), apply the chain rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). Substitute \(\frac{dy}{du} = -3u^{-4}\) and \(\frac{du}{dx} = \sec^2 x + \cos x\).

Step 5: Substitute the inner function back in

Replace \(u\) in \(-3u^{-4}\) with \(\tan x + \sin x\). This gives \(\frac{dy}{dx} = -3(\tan x + \sin x)^{-4}(\sec^2 x + \cos x)\).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. A composite function is when one function is placed inside another. In our example, we have an outer function and an inner function. The inner function is:

• \(u = \tan x + \sin x\)

The outer function is:

• \(y = u^{-3}\).

The chain rule states that to find the derivative of a composite function, you need to take the derivative of the outer function first while treating the inner function as a variable, and then multiply it by the derivative of the inner function. This can be expressed as \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \), where:

• \(\frac{dy}{du}\) is the derivative of the outer function with respect to \(u\),
• \(\frac{du}{dx}\) is the derivative of the inner function with respect to \(x\).

By substituting the specific derivatives into the formula, you can find the overall rate of change for the composite function. This powerful technique allows us to tackle more complex differentiation problems with ease.

Power Rule

The power rule is a basic technique in differentiation used to find the derivative of functions in the form \(y = x^n\), where \(n\) is a real number. According to the power rule, the derivative of \(y = x^n\) with respect to \(x\) is: \[ \frac{d}{dx} x^n = nx^{n-1} \] In the current exercise, this rule is applied to the outer function, which is \(y = u^{-3}\). Using the power rule, we find its derivative with respect to \(u\) as follows:

• \(\frac{dy}{du} = -3u^{-4}\).

This shows how the power rule simplifies the process of differentiation. By following the rule's formula, complex expressions can be broken down into manageable parts. In calculus, mastering the power rule is essential as it's a widely applicable shortcut for differentiating polynomials and powers.

Trigonometric Functions

Trigonometric functions play a critical role in calculus because they often appear in the context of waveforms, circles, and other periodic phenomena. In this exercise, the inner function \(u = \tan x + \sin x\) includes two commonly used trigonometric functions: tangent and sine.

• The derivative of \(\tan x\) is \(\sec^2 x\). This derivative arises because of the unique relationship between tangent and secant functions.
• The derivative of \(\sin x\) is \(\cos x\), resulting from the fundamental properties of sine and cosine relationships.

To differentiate the inner function, you simply add the derivatives of \(\tan x\) and \(\sin x\) together, resulting in \(\frac{du}{dx} = \sec^2 x + \cos x\). Understanding how to differentiate these trigonometric functions is crucial for solving many calculus problems involving waves and oscillations.