Question

Calculate the derivative of the following functions. $$y=\left(1-e^{-0.05 x}\right)^{-1}$$

Short answer

Answer: The derivative of the function is \(\frac{dy}{dx} = -0.05 e^{-0.05 x}\left(1-e^{-0.05x}\right)^{-2}\).

Step-by-step solution

Identify the outer and inner functions

In this problem, the outer function \(f(u) = u^{-1}\) and the inner function \(g(x) = 1 - e^{-0.05x}\). Now, we will find the derivatives of these two functions.

Find the derivative of the outer function

First, let's find the derivative of the outer function. For \(f(u) = u^{-1}\), according to the power rule, the derivative is: $$f'(u) = -u^{-2}$$

Find the derivative of the inner function

Now, let's find the derivative of the inner function. For \(g(x) = 1 - e^{-0.05x}\), the derivative can be found using the chain rule for \(-0.05x\) inside the exponential function: $$g'(x) = - (-0.05) e^{-0.05 x} = 0.05 e^{-0.05 x}$$

Apply the chain rule

Now that we have the derivatives of the outer and inner functions, we will apply the chain rule to find the derivative of the composite function: $$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$ Substitute the derivatives and functions into the chain rule: $$\frac{dy}{dx} = f'(1-e^{-0.05x}) \cdot 0.05 e^{-0.05 x}$$ Plug in \(f'(u) = -u^{-2}\) and simplify: $$\frac{dy}{dx} = -\left(1-e^{-0.05x}\right)^{-2} \cdot 0.05 e^{-0.05 x}$$

Write the final answer

The derivative of the given function is: $$\frac{dy}{dx} =-0.05 e^{-0.05 x}\left(1-e^{-0.05x}\right)^{-2}$$

Key concepts

Chain Rule

Understanding the chain rule is crucial in calculus, especially when dealing with composite functions, where one function is inside of another. It allows us to differentiate complex expressions by breaking them down into simpler ones.

When we have a composite function, we can think of it as a combination of two functions, say, an outer function, denoted by \(f\), and an inner function, denoted by \(g\). The derivative of the composite function \(f(g(x))\) is found by taking the derivative of the outer function with respect to the inner function and then multiplying it by the derivative of the inner function with respect to \(x\). In formula form, the chain rule states that \[\frac{d}{dx}f(g(x)) = f'(g(x)) \times g'(x)\].

To illustrate, consider the function \(y = (1-e^{-0.05x})^{-1}\). Here, the outer function is \(f(u) = u^{-1}\) and the inner function is \(g(x) = 1 - e^{-0.05x}\). According to the chain rule, we first find the derivatives of both functions separately, and then multiply them to get the derivative of the composite function. This process simplifies the differentiation task and is widely applicable in many areas where composite functions are present.

Exponential Functions

Exponential functions are a class of mathematical functions characterized by an expression of the form \(b^x\), where \(b\) is a positive constant base and \(x\) is any real number exponent. In the natural exponential function, the base is \(e\), which is an irrational number approximately equal to 2.71828, and holds a special place in calculus due to its unique properties in relation to the rate of change.

When it comes to finding the derivative of exponential functions, they have a remarkable trait: the derivative of an exponential function \(e^x\) is itself, \(e^x\). However, if the exponent is a function of \(x\), like \(e^{-0.05x}\), we must involve the chain rule. In the given function \(y = (1-e^{-0.05x})^{-1}\), the exponential part \(e^{-0.05x}\) is crucial. Its derivative is \(0.05 e^{-0.05x}\) due to the constant multiple rule applied in conjunction with the chain rule.

The exponential function's continuous growth or decay models many natural phenomena such as population growth, radioactive decay, and even financial investments, making its understanding important in both theory and real-world applications.

Power Rule

The power rule is a fundamental technique used to differentiate functions of the form \(x^n\), where \(n\) is any real number. This rule states that the derivative of \(x^n\) is \(nx^{n-1}\). It provides a quick and straightforward way to find the derivative of power functions without the need for limits.

Applying the power rule to the outer function \(f(u) = u^{-1}\) in the original exercise leads to the derivative \(f'(u) = -u^{-2}\), which is obtained by multiplying the exponent \(n=-1\) by the function and decreasing the exponent by one. The power rule, often used in conjunction with the chain rule, is an essential tool for calculating the derivatives of polynomial functions and, more broadly, any function where variables are raised to powers.

In practice, the power rule not only saves time but also simplifies complex calculations, making it easier for students and professionals alike to work efficiently with powers in differentiation.