Problem 69 - Calculus for Biology and Medicine

Question

Find the derivative of $$ f(x)=\sin ^{2}\left(x^{2}-1\right) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 2x \sin(2x^2 - 2) \).

Step 1: Identify the Composite Function

The function given is a composite function of the form \( f(x) = g(h(x)) \) where \( g(x) = \sin^2(x) \) and \( h(x) = x^2-1 \). We need to find \( f'(x) \) using the chain rule.

Step 2: Differentiate the Outer Function

The outer function \( g(x) = \sin^2(x) \) can be rewritten as \( (\sin(x))^2 \). The derivative of \( g(x) \) with respect to \( x \) using the chain rule and power rule is \( 2\sin(x)\cos(x) = \sin(2x) \).

Step 3: Differentiate the Inner Function

The inner function \( h(x) = x^2 - 1 \) is a simple polynomial. The derivative of \( h(x) \) with respect to \( x \) is \( h'(x) = 2x \).

Step 4: Apply the Chain Rule

The chain rule states that \( f'(x) = g'(h(x)) \times h'(x) \). Substituting the derivatives we found: \( g'(h(x)) = \sin(2(x^2-1)) \) and \( h'(x) = 2x \). Thus, \( f'(x) = \sin(2(x^2-1)) \times 2x \).

Step 5: Simplify the Expression

Distributing and simplifying, the derivative is \( f'(x) = 2x \sin(2x^2 - 2) \).

Key Concepts

Composite Function

A composite function is a combination of two functions where one function is applied to the result of another. For the given problem, the function can be expressed as \( f(x) = (\sin(x))^2 \), where \( x \) is replaced by another function, \( x^2 - 1 \). Hence, it takes the form \( f(x) = g(h(x)) \), with \( g(x) = \sin^2(x) \) and \( h(x) = x^2 - 1 \). Composite functions are like layering functions together.

• Identify the two functions separately: the outer function and the inner function.
• In this exercise, \( g(h(x)) \) shows the trigonometric function is applied to the polynomial.

Recognizing these layers helps us use the chain rule in the next steps.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It allows us to find the derivative of a composition by differentiating each layer of the function. The key idea is to take the derivative of the outer function and multiply it by the derivative of the inner function.

• The rule can be expressed as: if \( f(x) = g(h(x)) \), then the derivative \( f'(x) = g'(h(x)) \times h'(x) \).

This means we first find how the outer function changes with the inner function held constant, then how the inner function changes with respect to \( x \), and multiply them together. This gives us the overall rate of change of the composite function. In our problem, it simplifies the task of handling the derivative of \( \sin^2(x^2 - 1) \).

Derivative

Derivatives measure how a function changes as its input changes — essentially, the rate of change or slope of the function at any given point. For polynomials like \( h(x) = x^2 - 1 \), taking the derivative is straightforward:

• The derivative of \( x^2 \) is \( 2x \), and the derivative of a constant is 0.

Applying this to \( h(x) \), we find \( h'(x) = 2x \). In our exercise, we combine this with the derivative of the outer function using the chain rule. The outer function, written as \( \sin^2(x) \), differentiates to \( 2\sin(x)\cos(x) \), or \( \sin(2x) \) using the double angle identity. This creates a product of these insights: \( 2x \cdot \sin(2(x^2 - 1)) \).

Trigonometric Functions

Trigonometric functions are key in this exercise. They include functions like sine and cosine, which are periodic and describe waves. The function \( \sin(x) \) specifically describes the y-coordinate of a point on the unit circle. Here, we use the sine function raised to a power, as in \( \sin^2(x) \).

• Remember, \( \sin^2(x) \) is equivalent to \( (\sin(x))^2 \) and follows both power and trigonometric derivative rules.
• The derivative of the sine function is \( \cos(x) \).

When differentiating \( \sin^2(x) \), applying the power rule alongside the derivative of sine produces \( 2\sin(x)\cos(x) \), known as \( \sin(2x) \) by a trigonometric identity. This simplifies the differentiation process in composite functions.