Problem 59 - Calculus and its applications

Question

Use the Chain Rule, implicit differentiation, and other techniques to differentiate each function given. $$ y=2^{x^{4}} $$

Step-by-Step Solution

Answer

The derivative of \( y = 2^{x^4} \) is \( 4x^3 \cdot 2^{x^4} \ln(2) \).

Step 1: Identify the Functions

The given function is \( y = 2^{x^4} \). Here, we recognize an exponential function of the form \( a^{u(x)} \), where \( a = 2 \) and \( u(x) = x^4 \).

Step 2: Apply the Chain Rule for Exponential Functions

For differentiating \( a^{u(x)} \), use the formula \( \frac{d}{dx} [a^{u(x)}] = a^{u(x)} \ln(a) \cdot \frac{du}{dx} \).

Step 3: Differentiate the Inner Function

Now, we differentiate the inner function \( u(x) = x^4 \). The derivative is given by \( \frac{du}{dx} = 4x^3 \) using power rule for differentiation.

Step 4: Combine the Derivatives

Recall the derivative found in Step 2, \( \frac{d}{dx} [2^{x^4}] = 2^{x^4} \ln(2) \cdot \frac{du}{dx} \). Now substitute \( \frac{du}{dx} = 4x^3 \) from Step 3.This gives \( \frac{d}{dx} [2^{x^4}] = 2^{x^4} \ln(2) \cdot 4x^3 \).

Step 5: Simplify the Derivative

Simplify the expression obtained:\( \frac{d}{dx} [2^{x^4}] = 4x^3 \cdot 2^{x^4} \ln(2) \).

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used when it's challenging to solve an equation for one variable explicitly before differentiating. This method is particularly handy when dealing with equations where the variable we wish to differentiate is tangled with others and is not easily isolated. Here’s what you need to remember:

• When we have an equation involving both \(x\) and \(y\) that defines \(y\) implicitly as a function of \(x\), we differentiate both sides of the equation with respect to \(x\).
• In doing so, treat \(y\) as a function of \(x\), meaning each time you differentiate terms involving \(y\), multiply with \(\frac{dy}{dx}\).

Though not directly used in the original exercise, implicit differentiation becomes important when exponential functions are hidden within other complex expressions, or when a differentiation problem involves multiple related variables.

Exponential Functions

Exponential functions are a class of mathematical functions involving exponents. They have the form \(a^{x}\), where \(a\) is a constant and \(x\) is the variable.

• In this exercise, the function is \(y = 2^{x^4}\), which is a perfect example of an exponential function.
• Exponential functions grow very rapidly, which is why they are useful in modeling scenarios involving swift change, like population growth or radioactive decay.

When differentiating exponential functions of the form \(a^{u(x)}\), we employ the chain rule to efficiently break down the problem.

• We start by recognizing the base \(a\) and the inner function \(u(x)\).
• The key formula here is: \(\frac{d}{dx}[a^{u(x)}] = a^{u(x)} \ln(a) \cdot \frac{du}{dx}\).

This formula dictates that we first take the natural logarithm, \(\ln(a)\), which scales the derivative appropriately, and then multiply by the derivative of the exponent \(u(x)\).

Power Rule for Differentiation

The power rule is a fundamental technique in differentiation. It helps simplify the process of finding the derivative of powers of \(x\).

For any term \(x^{n}\), where \(n\) is a real number, the power rule states:

• The derivative \(\frac{d}{dx}[x^{n}] = nx^{n-1}\).

This rule was crucial in our step-by-step solution when differentiating \(u(x) = x^4\) from the given exponential function. We applied the power rule as follows:

• For \(x^{4}\), the derivative is calculated to be \(4x^{3}\), indicating that the power of \(x\) decreases by one and is multiplied by the original power.

Understanding and applying the power rule makes solving more complex derivatives, like those involving the chain rule, much easier and more intuitive. It's a powerful tool in your calculus toolkit, especially when dealing with polynomial expressions within exponential and other composite functions.