Understanding and Applying the Quotient Rule in Calculus Differentiation

quotient rule

The quotient rule is a formula used in calculus to differentiate a function that is the ratio of two other functions

The quotient rule is a formula used in calculus to differentiate a function that is the ratio of two other functions. It can be stated as follows:

If you have a function of the form f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions, then the derivative of f(x) is given by:

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

In simple terms, to apply the quotient rule, you take the derivative of the numerator, multiply it by the denominator, then subtract the derivative of the denominator multiplied by the numerator. Finally, you divide the whole expression by the square of the denominator.

Let’s look at an example to put the quotient rule into practice:

Example:

Find the derivative of f(x) = (3x^2 – 2x + 1) / (x – 1)

Solution:

To find the derivative of f(x) using the quotient rule, we need to determine the derivatives of both the numerator and the denominator.

Numerator:
g(x) = 3x^2 – 2x + 1
g'(x) = d/dx (3x^2 – 2x + 1) = 6x – 2

Denominator:
h(x) = x – 1
h'(x) = d/dx (x – 1) = 1

Now, we can apply the quotient rule formula:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2
f'(x) = ((6x – 2) * (x – 1) – (3x^2 – 2x + 1) * 1) / (x – 1)^2
f'(x) = (6x^2 – 8x + 2) / (x – 1)^2

So, the derivative of f(x) is f'(x) = (6x^2 – 8x + 2) / (x – 1)^2.

This is the general process of using the quotient rule. Remember to simplify the final expression if possible. Practice and familiarity with differentiating functions using the quotient rule will help you become more efficient with solving calculus problems involving ratios of functions.

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