Calculate the derivative of the following functions.
y = (1 - e^0.05x)^-1

Verified step by step guidance

1. Step 1: Identify the function y = (1 - e^{0.05x})^{-1} as a composition of functions, where the outer function is u^{-1} and the inner function is u = 1 - e^{0.05x}.
2. Step 2: Apply the chain rule for differentiation, which states that the derivative of a composite function y = f(g(x)) is given by y' = f'(g(x)) * g'(x).
3. Step 3: Differentiate the outer function u^{-1} with respect to u, which gives -u^{-2}.
4. Step 4: Differentiate the inner function u = 1 - e^{0.05x} with respect to x, which gives -0.05e^{0.05x}.
5. Step 5: Combine the results from Steps 3 and 4 using the chain rule: y' = -u^{-2} * (-0.05e^{0.05x}), and substitute u = 1 - e^{0.05x} back into the expression.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative can be computed using various rules, such as the power rule, product rule, quotient rule, and chain rule, depending on the form of the function.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), the derivative dy/dx can be found by multiplying the derivative of the outer function f with respect to g by the derivative of the inner function g with respect to x. This rule is essential when differentiating functions that are nested within each other.

Exponential Functions

Exponential functions are mathematical functions of the form y = a * e^(bx), where e is the base of the natural logarithm, and a and b are constants. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus. The derivative of an exponential function is unique in that it is proportional to the function itself, making it crucial for solving problems involving growth rates.