Problem 3 - Calculus

Question

Find the derivatives of the functions $$ \left(e^{x}\right)^{8} $$

Step-by-Step Solution

Answer

The derivative of \((e^x)^8\) is \(8e^{8x}\).

Step 1: Rewrite the Function

First, rewrite the function \( (e^x)^8 \) using the laws of exponents. This is equivalent to \( e^{8x} \) because \( (a^m)^n = a^{mn} \).

Step 2: Use the Chain Rule

Differentiate the function \( e^{8x} \) using the chain rule. The chain rule states that if you have a function \( f(g(x)) \), the derivative is \( f'(g(x)) \cdot g'(x) \). Here, \( f(u) = e^u \) and \( g(x) = 8x \).

Step 3: Differentiate the Outer Function

The derivative of \( f(u) = e^u \) with respect to \( u \) is \( e^u \). Therefore, the derivative of \( e^{8x} \) with respect to \( 8x \) is \( e^{8x} \).

Step 4: Differentiate the Inner Function

The derivative of the inner function \( g(x) = 8x \) with respect to \( x \) is \( 8 \).

Step 5: Combine Derivatives

Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function: \( e^{8x} \times 8 \).

Step 6: Simplify the Expression

Simplifying the expression gives \( 8e^{8x} \).

Key Concepts

Exponent Laws

Exponent laws are fundamental in simplifying expressions and finding derivatives of complex exponential functions. These laws provide us rules like \((a^m)^n = a^{mn}\), which allows us to handle powers raised to other powers.
For instance, in the given problem, the exponential expression \((e^x)^8\) is rewritten using exponent laws.

• The function \((e^x)^8\) can be simplified to \(e^{8x}\), making further calculations easier.


• This transformation uses the property that the exponents multiply together: \((a^m)^n = a^{mn}\).

Understanding exponent laws can simplify many mathematical operations especially when dealing with derivatives, making them crucial when tackling calculus problems.

Chain Rule

The chain rule is an essential tool in differentiation, especially for composite functions. It provides a systematic way to differentiate functions composed of other functions.
When you have a function like \(f(g(x))\), the chain rule states that the derivative is \(f'(g(x)) \cdot g'(x)\).

• In our example, the function is \(e^{8x}\), represented as \(f(u)\) with \(u = 8x\).


• The chain rule helps us to multiply the derivative of the outer function by the derivative of the inner function.

This method simplifies the process of finding derivatives, which is crucial for handling more complex variable expressions in calculus.

Outer Function Derivation

Deriving the outer function is a key part of using the chain rule effectively. The outer function in the problem is \(f(u) = e^u\).
The derivative here is straightforward because the derivative of \(e^u\) with respect to \(u\) is simply \(e^u\).

• This simplicity makes exponential functions like \(e^u\) very convenient to differentiate.


• Thus, for our problem, we find that the derivative of \(e^{8x}\) with respect to its inner component is \(e^{8x}\).

Understanding the differentiation of outer functions allows you to break down a problem systematically and applies widely, especially with exponential functions in calculus.

Inner Function Derivation

To complete our use of the chain rule, we need to differentiate the inner function. Here, the inner function is \(g(x) = 8x\).
Deriving this function is simple, as it is a linear expression. The derivative of \(8x\) with respect to \(x\) is just \(8\).

• This step provides the scaling factor needed in the chain rule multiplication.


• Understanding inner function derivation aids in logically reconstructing complex function derivatives.

By combining this result with the previously derived outer function, we fully apply the chain rule to find the overall derivative. This sequential process is essential for tackling a variety of function compositions in differential calculus.