Problem 40 - Calculus - AP Edition

Question

Find \(d y / d x\) $$y=\left[1+\sin ^{3}\left(x^{5}\right)\right]^{12}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = 180 x^4 \left[ 1 + \sin^3(x^5) \right]^{11} \sin^2(x^5) \cos(x^5) \).

Step 1: Identify the Outer Function

The function is given by \( y = \left[1+\sin ^{3}\left(x^{5}\right)\right]^{12} \). The outermost function is a power function with exponent 12, i.e., \( \left(u\right)^{12} \), where \( u = 1+\sin^{3}(x^5) \).

Step 2: Differentiate the Outer Function

Differentiate \( y \) with respect to \( u \) using the power rule: \( \frac{dy}{du} = 12\cdot \left(u\right)^{11} \). Substitute back \( u = 1+\sin^{3}(x^5) \), so \( \frac{dy}{du} = 12 \left[ 1 + \sin^3(x^5) \right]^{11} \).

Step 3: Identify the Middle Function

The middle part of the function inside is \( 1+\sin^3(x^5) \), where the function \( u = v^3 + 1 \) and \( v = \sin(x^5) \).

Step 4: Differentiate the Middle Function

Differentiate \( u = v^3 + 1 \) with respect to \( v \) to get \( \frac{du}{dv} = 3v^2 \). Substitute back \( v = \sin(x^5) \), so \( \frac{du}{dv} = 3 \sin^2(x^5) \).

Step 5: Identify the Innermost Function

The innermost function is \( v = \sin(x^5) \). Differentiate \( v \) with respect to \( x \).

Step 6: Differentiate the Innermost Function

Differentiate \( v = \sin(x^5) \) using the chain rule. \( \frac{dv}{dx} = \cos(x^5) \cdot 5x^4 \) because the derivative of \( \sin(t) \) is \( \cos(t) \) and of \( x^5 \) is \( 5x^4 \).

Step 7: Apply the Chain Rule

Combine all the derivatives using the chain rule: \( \frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dv}\cdot\frac{dv}{dx} \). Substitute in the values obtained from previous steps: \( \frac{dy}{dx} = 12 \left[ 1 + \sin^3(x^5) \right]^{11} \times 3 \sin^2(x^5) \times \cos(x^5) \times 5x^4 \).

Step 8: Simplify the Expression

Simplify the expression for \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = 180 x^4 \left[ 1 + \sin^3(x^5) \right]^{11} \sin^2(x^5) \cos(x^5) \].

Key Concepts

Differentiation

Differentiation is a core concept in calculus that helps us understand how a function changes at any given point. It involves finding the derivative, which represents the rate of change. In the context of our exercise, we are tasked with differentiating a composite function using the chain rule.When we have a composite function, like the one presented in this exercise, differentiation isn't always straightforward. This is where the chain rule comes into play, allowing us to differentiate functions that are made up of multiple layers. The chain rule states that to differentiate a composite function, denoted as \( y = f(g(x)) \), we multiply the derivative of the outer function by the derivative of the inner function.In our exercise, the structure is more complex due to multiple layers (or functions within functions). By identifying each layer, starting from the outermost to the innermost, we can simplify and successfully differentiate the entire function.

Power Rule

The power rule is one of the most commonly used rules in differentiation. It applies to any function of the form \( x^n \), where \( n \) is a constant. The rule simplifies the process of finding derivatives, stating that if \( y = x^n \), then the derivative \( \frac{dy}{dx} \) is \( nx^{n-1} \).In the given exercise, the power rule is applied when differentiating the outer function \( y = \, [1 + \sin^3(x^5)]^{12} \). This is recognized as a power function with \( u^{12} \), where \( u = 1 + \sin^3(x^5) \). Applying the power rule, the derivative is \( 12u^{11} \).This is a fundamental rule that enables us to handle functions raised to any power efficiently, especially those nested within more complex expressions.

Trigonometric Functions

Trigonometric functions, such as \( \sin \), \( \cos \), and \( \tan \), are functions of angles and are crucial in calculus due to their periodic nature and widespread applications. In differentiation, each trigonometric function has its own derivative:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• The derivative of \( \tan(x) \) is \( \sec^2(x) \).

For our exercise, the function \( \sin(x^5) \) is present and needs differentiation in the context of a product \( \sin^3(x^5) \). Using the chain rule, we identify \( \sin(x^5) \) as the innermost function. Its derivative is found by applying the derivative of \( \sin \), followed by considering the inner function derivative, resulting in \( \cos(x^5) \cdot 5x^4 \).Understanding how to differentiate trigonometric functions is essential for tackling a wide variety of problems involving oscillatory behavior.