9–61. Evaluate and simplify y'.


y = e^2θ

Verified step by step guidance

1. Identify the given function: y = e^(2θ). We need to find the derivative y' with respect to θ.
2. Recognize that the function y = e^(2θ) is an exponential function where the exponent is a function of θ. This requires the use of the chain rule for differentiation.
3. Apply the chain rule: If y = e^(u) where u is a function of θ, then the derivative y' = e^(u) * (du/dθ). Here, u = 2θ.
4. Differentiate u = 2θ with respect to θ to find du/dθ. The derivative of 2θ with respect to θ is 2.
5. Substitute back into the chain rule formula: y' = e^(2θ) * 2. Simplify the expression to get the final form of the derivative.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. In this case, we need to differentiate the function y = e^(2θ) with respect to θ to find y'.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where e is the base of the natural logarithm. These functions exhibit rapid growth or decay and are characterized by their constant rate of change. Understanding the properties of exponential functions is crucial for differentiating them correctly.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential when differentiating functions like y = e^(2θ), where the exponent is itself a function of θ.