How to Find the Derivative of a Quotient of Two Functions – The Quotient Rule Explained

d/dx [f(x)/g(x)] =

To find the derivative of the quotient of two functions, f(x) and g(x), with respect to x, we can use the quotient rule

To find the derivative of the quotient of two functions, f(x) and g(x), with respect to x, we can use the quotient rule.

The quotient rule states that if f(x) and g(x) are differentiable functions, then the derivative of f(x)/g(x) with respect to x, denoted as d/dx [f(x)/g(x)], can be computed as:

d/dx [f(x)/g(x)] = (g(x) * f'(x) – f(x) * g'(x))/[g(x)]^2

Here, f'(x) represents the derivative of f(x) with respect to x, and g'(x) represents the derivative of g(x) with respect to x.

To apply the quotient rule, we need to differentiate each function separately, and then substitute them into the formula. Let’s break it down step by step.

1. Differentiate f(x) to find f'(x).
2. Differentiate g(x) to find g'(x).
3. Substitute f(x), f'(x), g(x), and g'(x) into the quotient rule formula.
4. Simplify the expression if possible.

Let’s use an example to illustrate this process:

Suppose we have f(x) = 3x^2 and g(x) = 4x + 1. We want to find d/dx [f(x)/g(x)].

1. Differentiating f(x):
f'(x) = 6x

2. Differentiating g(x):
g'(x) = 4

3. Applying the quotient rule:
d/dx [f(x)/g(x)] = (g(x) * f'(x) – f(x) * g'(x))/[g(x)]^2
= ((4x + 1) * 6x – 3x^2 * 4)/[(4x + 1)]^2
= (24x^2 + 6x – 12x^2)/(16x^2 + 8x + 1)
= (12x^2 + 6x)/(16x^2 + 8x + 1)

So, d/dx [f(x)/g(x)] = (12x^2 + 6x)/(16x^2 + 8x + 1).

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