Problem 24 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=-3 \cos ^{2}\left(3 x^{2}-4\right) $$

Step-by-Step Solution

Answer

The derivative is \( 18x \sin(6x^2 - 8) \).

Step 1: Identify the function composition

The function given is a composition of a cosine squared function and a quadratic function. Rewriting it using another variable, let \( u = 3x^2 - 4 \), then the function becomes \( f(x) = -3 \cos^2(u) \). Differentiate using the chain rule.

Step 2: Differentiate the outside function

Using the chain rule, differentiate \( -3 \cos^2(u) \) with respect to \( u \). Since \( -3 \cos^2(u) \) can be rewritten as \( -3 \times (\cos(u))^2 \), use the derivative of the power and the chain rule: \( \frac{d}{du}(-3(\cos(u))^2) = -3 \times 2\cos(u)(-\sin(u)) \). Simplifying this gives \( 6 \cos(u) \sin(u) \).

Step 3: Differentiate the inside function

Differentiate the inner function \( u = 3x^2 - 4 \) with respect to \( x \). This derivative is \( \frac{du}{dx} = 6x \), using the power rule.

Step 4: Combine results using the chain rule

Combine the derivatives from Step 2 and Step 3 using the chain rule: \( \frac{df}{dx} = 6 \cos(u) \sin(u) \times 6x \). This results in \( 36x \cos(u) \sin(u) \).

Step 5: Simplify using trigonometric identity

Use the identity \( \sin(2u) = 2\sin(u)\cos(u) \) to simplify the expression. Thus, \( 36x \cos(u) \sin(u) = 18x \sin(2u) \).

Step 6: Substitute back the original expression for u

Recall that \( u = 3x^2 - 4 \). Substitute back to get \( \frac{df}{dx} = 18x \sin(2 \times (3x^2 - 4)) \). Therefore, the derivative is \( 18x \sin(6x^2 - 8) \).

Key Concepts

Derivative

In calculus, a derivative represents the rate of change of a function with respect to a variable. It tells us how a function behaves as its input changes. For example, if we have a function like \( f(x) = ax^2 + bx + c \), the derivative \( f'(x) \) would help determine how the value of \( f \) changes as \( x \) changes. Derivatives are essential in understanding the behavior of functions:

• They help to find the slope of the tangent line to the curve at any point.
• They are used to find maximum and minimum points on a graph, which are crucial in optimization problems.
• They provide insights into the increasing or decreasing nature of functions.

To find the derivative, basic rules like the power rule, product rule, quotient rule, and chain rule are applied depending on the function's composition. In this case, we find the derivative of a function combining trigonometric and polynomial components using these rules.

Chain Rule

The chain rule is a formula used to compute the derivative of a composition of functions. When dealing with complex functions, like a function within another function, the chain rule simplifies the differentiation process.For the function provided, \( f(x) = -3 \cos^2(3x^2 - 4) \), the chain rule allows us to take the derivative by identifying the inner and outer functions.

• The inner function is \( u = 3x^2 - 4 \).
• The outer function is \( v = -3 \cos^2(u) \).

To apply the chain rule: 1. Differentiate the outer function with respect to \( u \). This involves using the derivative of a power function and the trigonometric function.2. Differentiate the inner function with respect to \( x \).3. Multiply these derivatives together to get the full derivative of the original function concerning \( x \), combining the results appropriately, as the chain rule prescribes.This process helps break down complex functions into manageable parts, making it easier to calculate derivatives.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides are defined. These identities are powerful tools in calculus, particularly when simplifying derivatives of trigonometric functions.In the exercise, the identity \( \sin(2u) = 2 \sin(u) \cos(u) \) is crucial. This identity helps simplify expressions involving products of sine and cosine:

• The original derivative expression \( 36x \cos(u) \sin(u) \) uses this identity to become \( 18x \sin(2u) \).

Such simplifications:- Make derivatives more manageable and easier to interpret.- Save time in calculations, especially in more complex problems.- Provide a clearer understanding of the trigonometric behavior in compositions of functions.Understanding how to apply these identities is vital for working efficiently with trigonometric derivatives, making the solutions neat and elegant.