Problem 67 - Calculus for Biology and Medicine

Question

Find the derivative of $$ f(x)=\sin \sqrt{3 x^{3}+3 x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \cos(\sqrt{3x^3 + 3x}) \cdot \frac{9x^2 + 3}{2\sqrt{3x^3 + 3x}} \).

Step 1: Identify the Components of the Function

The function to be derived is given as \( f(x) = \sin(\sqrt{3x^3 + 3x}) \). This function is composed of a composition of a sine function and a square root function, which in turn contains a polynomial expression \( 3x^3 + 3x \).

Step 2: Apply the Chain Rule

The chain rule for derivatives states that \( \frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) \). Here \( f \) is \( \sin(u) \) and \( u = \sqrt{3x^3 + 3x} \), an inner function. We need to find \( f'(u) \) and \( u'(x) \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( f(u) = \sin(u) \) with respect to \( u \). The derivative is \( \cos(u) \). Thus, \( f'(u) = \cos(u) \).

Step 4: Differentiate the Inner Structure inside the Sine

Identify \( u = \sqrt{3x^3 + 3x} = (3x^3 + 3x)^{1/2} \). To differentiate, use the chain rule: \( u' = \frac{1}{2}(3x^3 + 3x)^{-1/2} \cdot \frac{d}{dx}(3x^3 + 3x) \).

Step 5: Differentiate the Polynomial Inside the Square Root

Differentiate \( 3x^3 + 3x \) with respect to \( x \). The derivative is \( 9x^2 + 3 \). Thus, the derivative of the square root component \( u' \) becomes \( \frac{1}{2}(3x^3 + 3x)^{-1/2} \cdot (9x^2 + 3) \).

Step 6: Apply the Chain Rule to Combine the Derivatives

We have \( f'(x) = \cos(\sqrt{3x^3 + 3x}) \cdot \frac{1}{2}(3x^3 + 3x)^{-1/2} \cdot (9x^2 + 3) \). Simplify to get the final answer.

Key Concepts

Chain Rule in Differentiation

When faced with composite functions, applying the chain rule becomes essential. The chain rule is a fundamental technique used to find the derivative of a function that comprises other functions within it. It's like peeling an onion, where each layer needs to be differentiated successfully.

The rule states: if you have a function that's composed of two functions, say \( f(g(x)) \), the derivative will be the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In mathematical terms:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) \)

To apply it effectively, first identify the outer and inner functions. Always start differentiating from the outside, progressing towards the innermost function. Each step builds on the previous one, culminating in the final derivative expression.

Understanding Composite Functions

Composite functions are functions made up of two or more simpler functions. In this case, you're looking at a sine function of a square root, which itself contains a polynomial.

The step-by-step construction of the original function may look like:

• Start with a polynomial, such as \( 3x^3 + 3x \).
• Take the square root of this polynomial to form \( \sqrt{3x^3 + 3x} \).
• Finally, apply the sine function to this square root to get \( \sin(\sqrt{3x^3 + 3x}) \).

Identifying the layers helps in applying the chain rule later since you can clearly see which parts need differentiating.

Differentiating Polynomials

Polynomial differentiation is straightforward. The derivative of a term \( ax^n \) is \( anx^{n-1} \). Simply multiply by the exponent and decrease the exponent by one.

Applying this to \( 3x^3 + 3x \):

• The derivative of \( 3x^3 \) becomes \( 9x^2 \).
• The derivative of \( 3x \) is \( 3 \).

Thus, the entire polynomial differentiates to \( 9x^2 + 3 \), forming a crucial part of the inner function's derivative calculation.

Role of the Sine Function in Differentiation

The sine function, \( \sin(x) \), is easy to differentiate. The derivative of \( \sin(u) \) with respect to its argument \( u \) is \( \cos(u) \).

In the context of our composite function, you're not directly differentiating \( x \), but rather an entire inner function which results from other operations. Thus, the sine component produces a \( \cos \) as an element of the derivative when applying the chain rule.

This element acts on the intermediate result of all underlying components being differentiated first.

Differentiating the Square Root

Differentiating a square root function like \( \sqrt{x} \) involves a specific rule. A square root can be rewritten as a power of one-half: \( x^{1/2} \).

Using the power rule here, if \( u = \sqrt{3x^3 + 3x} \), differentiate as follows:

• The derivative of \( u = (3x^3 + 3x)^{1/2} \) becomes \( \frac{1}{2}(3x^3 + 3x)^{-1/2} \).

This result is then multiplied by the derivative of the polynomial inside. Such steps form an essential part of differentiating the entire composite function, ensuring you chain back to all initial components accurately.