Problem 48 - Single Variable Calculus

Question

Find the derivative of the function. $$ y=\sec ^{3}\left(\frac{\sqrt{x}}{1+x}\right) $$

Step-by-Step Solution

Answer

The short answer is: \(\frac{dy}{dx} = \left(3\sec^2\left(\frac{\sqrt{x}}{1+x}\right) \sec\left(\frac{\sqrt{x}}{1+x}\right) \tan\left(\frac{\sqrt{x}}{1+x}\right) \right) \left( \frac{\frac{1}{2\sqrt{x}}(1+x) - \sqrt{x}}{(1+x)^2} \right)\)

Step 1: Identify the outer and inner functions

Inner function: \(u = \frac{\sqrt{x}}{1+x}\) Outer function: \(y = \sec^3(u)\) Now, we need to apply the chain rule. For this, we will first find the derivatives of the outer and inner functions. Step 2: Find the derivative of the outer function

Step 2: Find the derivative of the outer function

Outer function: \(y = \sec^3(u)\) Derivative of the outer function: \(\frac{dy}{du} = 3\sec^2(u)\sec(u)\tan(u)\) Step 3: Find the derivative of the inner function

Step 3: Find the derivative of the inner function

Inner function: \(u = \frac{\sqrt{x}}{1+x}\) Derivative of the inner function: \(\frac{du}{dx} = \frac{\frac{1}{2\sqrt{x}}(1+x) - \sqrt{x}}{(1+x)^2}\) Step 4: Apply the chain rule

Step 4: Apply the chain rule

Using the chain rule, we can find the derivative of the function as follows: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\) Now we just need to plug in the derivatives of the outer and inner functions we found in steps 2 and 3: \(\frac{dy}{dx} = \left(3\sec^2(u)\sec(u)\tan(u) \right) \left( \frac{\frac{1}{2\sqrt{x}}(1+x) - \sqrt{x}}{(1+x)^2} \right)\) Step 5: Substitute the inner function back into the derivative

Step 5: Substitute the inner function back into the derivative

Finally, substitute the inner function, \(u = \frac{\sqrt{x}}{1+x}\), back into the derivative: \(\frac{dy}{dx} = \left(3\sec^2\left(\frac{\sqrt{x}}{1+x}\right) \sec\left(\frac{\sqrt{x}}{1+x}\right) \tan\left(\frac{\sqrt{x}}{1+x}\right) \right) \left( \frac{\frac{1}{2\sqrt{x}}(1+x) - \sqrt{x}}{(1+x)^2} \right)\) This is the derivative of the given function.

Key Concepts

Derivative of Secant Function

Understanding the derivative of the secant function is crucial when tackling calculus problems that involve trigonometric functions. Secant, represented as \(\sec x\), is one of the six fundamental trigonometric functions and its derivative is derived using the quotient rule of derivatives, as \(\sec x\) can be defined as \(\frac{1}{\cos x}\). The derivative of \(\sec x\) is \(\sec x \tan x\).

This becomes slightly more complex when dealing with higher powers of secant, such as \(\sec^3 x\). For such cases, one generally applies the power rule in combination with the chain rule to find the derivative. For \(\sec^n x\), the derivative is \(n\sec^{n-1} x \tan x\), assuming 'n' is a positive integer. In the exercise provided, the secant function is raised to the third power, which necessitates using this approach in step 2 of the solution, yielding \(3\sec^2(u)\sec(u)\tan(u)\) as the derivative of the outer function.

Implicit Differentiation

Implicit differentiation is a technique used when a function is not explicitly solved for one variable. In such cases, the differentiation is performed with respect to one variable while treating other variables as functions of that variable. This practice applies to functions that are impossible or difficult to express explicitly.

Despite not being directly visible in the exercise's solution, the spirit of implicit differentiation underpins the process of differentiating the inner function in step 3. Here, we are treating \(u\) implicitly as a function of \(x\) while differentiating \(u = \frac{\sqrt{x}}{1+x}\). The derivative of the inner function \(\frac{du}{dx}\) involves applying the quotient rule, a concept closely related to implicit differentiation because we are finding the derivative of one expression with respect to another.

Inner and Outer Functions

The concepts of inner and outer functions are core to understanding the chain rule and its applications. An inner function is a function that is nested within another function, while the outer function is the one that contains the inner function. Identifying these parts is the first step in differentiating complex functions, as done in step 1 of the exercise.

In the given function \(y=\sec ^{3}\left(\frac{\sqrt{x}}{1+x}\right)\), \(u = \frac{\sqrt{x}}{1+x}\) serves as the inner function, and \(y = \sec^3(u)\) acts as the outer function. Observing the structure of the composition of functions allows us to apply the chain rule effectively to find the derivative of the composite function.

Applying the Chain Rule

The chain rule is a fundamental derivative rule that provides a method to differentiate composite functions. When a function \(y\) is composed of an outer function \(g\) applied to an inner function \(f\), the chain rule states that the derivative of \(y\) with respect to \(x\) is the product of the derivative of \(g\) with respect to \(f\) (\(g'(f(x))\)) and the derivative of \(f\) with respect to \(x\) (\(f'(x)\)).

In mathematical terms, if \(y = g(f(x))\), then \(\frac{dy}{dx} = g'(f(x)) \cdot f'(x)\). This is precisely what we did in step 4 of the solution, applying the chain rule to multiply the derivatives of the outer and inner functions (\(\frac{dy}{du}\) and \(\frac{du}{dx}\), respectively). The final step involves substituting the original inner function back into our differentiation formula, which provides us with the derivative with respect to \(x\) of the original composite function.