Problem 65 - Calculus Single Variable

Question

find \(f^{\prime}(x)\) for the given function \(f\). $$ f(x)=4 \sec ^{5}(\sqrt{7 x}) $$

Step-by-Step Solution

Answer

\( f'(x) = \frac{70}{\sqrt{7x}} \sec^4(\sqrt{7x}) \tan(\sqrt{7x}) \).

Step 1: Understanding the Chain Rule

To find the derivative of a composite function like \( f(x) = 4 \sec^5(\sqrt{7x}) \), we need to apply the chain rule. The chain rule states that if you have a function \( y = g(h(x)) \), then the derivative is \( \frac{dy}{dx} = g'(h(x)) \cdot h'(x) \). Here, \( g(u) = 4 \sec^5(u) \) and \( h(x) = \sqrt{7x} \).

Step 2: Taking the Outer Function Derivative

First, find the derivative of the outer function \( g(u) = 4 \sec^5(u) \). Using the power rule and the derivative of \( \sec(u) \), we have \( g'(u) = 4 \cdot 5 \sec^{4}(u) \tan(u) \). This simplifies to \( g'(u) = 20 \sec^4(u) \tan(u) \).

Step 3: Taking the Inner Function Derivative

Next, find the derivative of the inner function \( h(x) = \sqrt{7x} \). To differentiate \( \sqrt{7x} \), rewrite it as \( (7x)^{1/2} \). Using the chain rule inside this differentiation, \( h'(x) = \frac{1}{2}(7x)^{-1/2} \cdot 7 = \frac{7}{2\sqrt{7x}} \).

Step 4: Combining the Derivatives

Now apply the chain rule formula: \( f'(x) = g'(h(x)) \cdot h'(x) \). Substitute \( h(x) = \sqrt{7x} \), \( g'(u) = 20 \sec^4(u) \tan(u) \), and \( h'(x) = \frac{7}{2\sqrt{7x}} \) into the formula. Simplifying, \( f'(x) = 20 \sec^4(\sqrt{7x}) \tan(\sqrt{7x}) \cdot \frac{7}{2\sqrt{7x}} \), which simplifies to \( f'(x) = \frac{70}{\sqrt{7x}} \sec^4(\sqrt{7x}) \tan(\sqrt{7x}) \).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used for differentiating composite functions. A composite function is one where a function is applied to the result of another function. In simpler terms, you have a function nested inside another function. Let's imagine you're peeling an onion; the outer layer represents the outer function, and as you peel layer by layer (or differentiate), you work your way to the core. To apply the chain rule, consider a function like \( y = g(h(x)) \). To find its derivative,

• First, differentiate the outer function \( g \) with respect to its input, keeping the inside function as it is. This gives us \( g'(h(x)) \).
• Next, multiply this by the derivative of the inner function, \( h'(x) \).

So, the derivative using the chain rule is \( \frac{dy}{dx} = g'(h(x)) \cdot h'(x) \). This process is essential for complex functions like \( f(x) = 4 \sec^5(\sqrt{7x}) \), helping break down the differentiation step-by-step.

Composite Function

Composite functions are where the output of one function becomes the input of another. It's like an assembly line in a factory: one station processes a part before passing it to the next. In calculus, understanding composite functions is crucial for effectively using the chain rule. For example, if you have a function \( f(x) = 4 \sec^5(\sqrt{7x}) \), the function can be seen as composed of two parts:

• Inner function \( h(x) = \sqrt{7x} \)
• Outer function \( g(u) = 4 \sec^5(u) \)

Here, the inner function transforms \( x \), and the outer function takes the outcome of \( h(x) \) to produce the final result. By breaking down a composite function into these parts, you make derivatives more manageable by applying the chain rule efficiently.

Power Rule

The power rule is a handy tool for differentiating expressions where the variable is raised to a power. If you're tackling an expression like \( x^n \), the power rule states that its derivative is \( nx^{n-1} \). You might wonder why this matters in the solution. In our example, the function \( 4 \sec^5(\sqrt{7x}) \) uses the power rule in differentiating the outer function. Here's how:

• First, recognize that \( \sec^5(u) \) involves raising \( \sec(u) \) to the fifth power.
• Applying the power rule gives \( 5 \sec^4(u) \cdot \sec'(u) \).

Combining this with the constant multiplier, the derivative for the outer function \( g(u) \) is simplified with techniques like the power rule, which is intertwined with other rules for precise results.

Sec Function

The secant function, denoted as \( \sec(x) \), is one of the fundamental trigonometric functions. It is the reciprocal of the cosine function, meaning \( \sec(x) = \frac{1}{\cos(x)} \). Understanding its derivative is vital for calculus since it's often involved in more complex functions. When differentiating \( \sec(x) \), the derivative is \( \sec(x) \cdot \tan(x) \), resulting in:

• \( \frac{d}{dx}[\sec(x)] = \sec(x) \tan(x) \)

For a function like \( 4 \sec^5(\sqrt{7x}) \), the derivative of the secant function is integrated with other rules, creating a structured approach to derive its differentiation. Combining the secant function's derivative with strategies such as the power rule, and chain rule, you accomplish intricate differentiation tasks with clarity and precision.