How do I find the derivative of a fraction?

Finding the derivative of a fraction involves applying the quotient rule, which is a specific technique in calculus for differentiating functions that are expressed as the ratio of two other functions. Let’s break this down step by step to make it easier to understand.

The Quotient Rule Explained

The quotient rule states that if you have a function that can be expressed as the fraction of two differentiable functions, say \( f(x) = \frac{g(x)}{h(x)} \), then the derivative \( f'(x) \) can be found using the following formula:

f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}

Breaking Down the Formula

In this formula:

• g(x) is the numerator function.
• h(x) is the denominator function.
• g'(x) is the derivative of the numerator.
• h'(x) is the derivative of the denominator.

The numerator of the derivative is formed by multiplying the derivative of the numerator by the denominator and then subtracting the product of the numerator and the derivative of the denominator. The entire expression is then divided by the square of the denominator.

Example for Clarity

Let’s consider a specific example to illustrate this process. Suppose we want to find the derivative of the function:

f(x) = \frac{x^2 + 1}{x - 3}

Here, we identify:

• g(x) = x^2 + 1
• h(x) = x - 3

Next, we need to find the derivatives:

• g'(x) = 2x
• h'(x) = 1

Now, applying the quotient rule:

f'(x) = \frac{(2x)(x - 3) - (x^2 + 1)(1)}{(x - 3)^2}

Let’s simplify the numerator:

f'(x) = \frac{2x^2 - 6x - x^2 - 1}{(x - 3)^2}

This simplifies to:

f'(x) = \frac{x^2 - 6x - 1}{(x - 3)^2}

Final Thoughts

And there you have it! The derivative of the fraction \( \frac{x^2 + 1}{x - 3} \) is \( \frac{x^2 - 6x - 1}{(x - 3)^2} \). Remember, the key to using the quotient rule effectively is to carefully identify the numerator and denominator, find their derivatives, and then apply the formula correctly. With practice, this process will become second nature!