Problem 15 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative of \( f(x) = 3 \sin(x^2) \) is \( f'(x) = 6x \cos(x^2) \).

Step 1: Identify the Function Types

The function provided, \( f(x) = 3\sin(x^2) \), is a composition of functions, specifically a product of a constant and the sine function applied to \( x^2 \). It involves both the constant multiplication rule and the chain rule for differentiation.

Step 2: Apply the Chain Rule

First, apply the chain rule which states that the derivative of \( \sin(g(x)) \) is \( \cos(g(x)) \cdot g'(x) \). For this function, \( g(x) = x^2 \), so you will find the derivative of \( \sin(x^2) \) by calculating \( \cos(x^2) \cdot g'(x) \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( g(x) = x^2 \). Its derivative is \( g'(x) = 2x \). This gives us \( \cos(x^2) \cdot 2x \) for the derivative of \( \sin(x^2) \).

Step 4: Compute the Full Derivative

Multiply the constant 3 by the result from the chain rule application: the derivative of \( 3\sin(x^2) \) is \( 3 \cdot \cos(x^2) \cdot 2x \). Simplify this to get the final result: \( 6x \cos(x^2) \).

Step 5: Final Derivative Expression

Combine all steps to write the derivative of the function: \( f'(x) = 6x \cos(x^2) \). This uses the chain rule and multiplication by a constant.

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of a composite function. A composite function is essentially one function applied to the result of another, such as \( \sin(x^2) \). To differentiate composite functions, the chain rule states that you first need to take the derivative of the outer function while keeping the inner function unchanged. Then, multiply the result by the derivative of the inner function.

For example, with the function \( f(x) = \sin(x^2) \), \( \sin \) is the outer function and \( x^2 \) is the inner function. When applying the chain rule here:

• Differentiate \( \sin(u) \) for \( u = x^2 \), which results in \( \cos(x^2) \).
• Differentiating the inner function \( x^2 \) yields \( 2x \).

Thus, using the chain rule means multiplying these results, giving us \( \cos(x^2) \cdot 2x \). This process allows us to effectively handle complex differentiations without losing track of the function's layers.

Composition of Functions

In calculus, a composition of functions involves creating a new function by plugging one function into another. This is often written as \( f(g(x)) \), which means take the function \( g(x) \) and then apply \( f(x) \) to it. The exercise here deals with such a composition: \( f(x) = 3 \sin(x^2) \).

Understanding composition of functions is crucial because it helps to break down complex functions into more manageable parts. When you see a function like \( 3 \sin(x^2) \), you can decompose it into its components:

• The constant multiplier \( 3 \) simply scales the overall result.
• The function \( \sin(x^2) \) is the result of composing the sine function with the square function \( x^2 \).

This perspective helps in utilizing techniques like the chain rule because you can see exactly where to apply differentiation and multiplication rules. Once broken down, each component can be managed individually, and then their derivatives are combined to arrive at the final derivative expression.

Trigonometric Functions

When dealing with derivatives involving trigonometric functions like sine, cosine, and tangent, it's important to understand their basic properties and derivatives. Trigonometric functions relate to angles and lengths in a right-angled triangle, and are periodic in nature, which means they repeat themselves at regular intervals.

For the sine function, which is relevant here, its derivative is the cosine function. So, if you are given \( f(x) = \sin(x) \), the immediate derivative is \( f'(x) = \cos(x) \). This relationship is key when applying the derivative rules to trigonometric functions in calculus.

• Sine and cosine are the most commonly used trigonometric functions in calculus.
• The derivative of \( \sin(x) \) is \( \cos(x) \), and the derivative of \( \cos(x) \) is \(-\sin(x) \).

In the original exercise, the function we deal with is \( \sin(x^2) \). After using the chain rule, this trigonometric function uses its derivative property to transform into \( \cos(x^2) \). Understanding how to handle these functions within a calculus context is critical when calculating derivatives of more complex expressions.