Question

Derivatives Find and simplify the derivative of the following functions. $$g(x)=\frac{e^{x}}{x^{2}-1}$$

Short answer

Question: Find the derivative of the function \(g(x) = \frac{e^x}{x^2-1}\) and simplify the expression. Answer: The derivative of the given function is \(g'(x) = \frac{e^x(x^2 -1 -2x)}{(x^2-1)^2}\).

Step-by-step solution

Identify the functions f(x) and h(x)

In the given function, $$g(x) = \frac{e^x}{x^2-1},$$ we can identify the functions: \(f(x) = e^x\) and \(h(x) = x^2 - 1\)

Find the derivatives of f(x) and h(x)

Now, we will find the derivatives of the two functions: 1. The derivative of \(f(x) = e^x\) is \(f'(x) = e^x\) because the derivative of \(e^x\) is itself \(e^x\). 2. The derivative of \(h(x) = x^2 - 1\) is \(h'(x) = \frac{d}{dx}(x^2 - 1)\). Using the power rule, we get \(h'(x) = 2x\).

Apply the Quotient Rule

Now, we will apply the Quotient Rule which states that: \(g'(x) = \frac{f'(x)h(x) - f(x)h'(x)}{(h(x))^2}\) We know \(f'(x) = e^x\), \(h(x) = x^2 - 1\), and \(h'(x) = 2x\). Plugging these into the Quotient Rule formula, we get: $$g'(x) = \frac{(e^x)(x^2-1) - (e^x)(2x)}{(x^2-1)^2}$$

Simplify the derivative

We can simplify the expression by factoring out \(e^x\) from the numerator: $$g'(x) = \frac{e^x(x^2 -1 -2x)}{(x^2-1)^2}$$ This is the simplified form of the derivative of the given function, $$g'(x) = \frac{e^x(x^2 -1 -2x)}{(x^2-1)^2}.$$

Key concepts

Quotient Rule

The quotient rule is a special method in calculus for finding the derivative of a division of two functions. If you have a function that is the quotient of two other functions, such as \( g(x) = \frac{f(x)}{h(x)} \), you'll need to use this rule. It's particularly handy when each function has its own rules for differentiation. Here's how the quotient rule works: the derivative of a quotient \( g(x) \) is given by the formula:\[g'(x) = \frac{f'(x)h(x) - f(x)h'(x)}{(h(x))^2}\]To break it down:

• \( f'(x) \) and \( h'(x) \) are the derivatives of the top \( f(x) \) and bottom \( h(x) \) functions, respectively.
• \( h(x) \) squared \( (h(x))^2 \), is found in the denominator.
• The numerator is the subtraction of the product of \( f'(x)h(x) \) minus \( f(x)h'(x) \).

It's helpful to write each step out clearly and methodically, ensuring you substitute the correct derivative functions into the formula.

Exponential Function

Exponential functions are those where the variable appears as an exponent. The most common exponential function is \( e^x \), where \( e \) is Euler's number, approximately 2.71828. An amazing property of the exponential function \( e^x \) is that it is its own derivative. This means that if you differentiate \( e^x \), you get \( e^x \). This makes it incredibly simple to handle when differentiating.For example, let's say \( f(x) = e^x \):

• The derivative \( f'(x) = e^x \).
• This property holds true no matter how you transform the function. For instance, when used in operations like addition or scaling, you still derive the same way.

This feature simplifies calculations considerably and makes the exponential function a cornerstone in calculus.

Power Rule

The power rule is one of the most basic rules for differentiation. It is extremely useful when you need to find the derivative of a function of the form \( x^n \), where \( n \) is any real number. The rule states that the derivative of \( x^n \) is:\[\frac{d}{dx}(x^n) = nx^{n-1}\]Here's how it works:

• Take the exponent \( n \) and multiply it by the coefficient of \( x \).
• Reduce the exponent by one to form the new power for \( x \).

For instance, if you are given \( x^2 \), applying the power rule gives:

• \( n = 2 \), so the derivative is \( 2x^{2-1} = 2x \).

This rule is straightforward and becomes second nature with practice, especially as you apply it to more complex problems, such as combining it with the quotient rule as in our case.