Quotient Rule
The quotient rule is a rule in calculus that helps us find the derivative of a quotient of two functions
The quotient rule is a rule in calculus that helps us find the derivative of a quotient of two functions. It is used when we have a function that is in the form of f(x) = g(x) / h(x), where g(x) and h(x) are two differentiable functions.
The formula for the quotient rule is:
f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2
where f'(x) is the derivative of f(x), g'(x) is the derivative of g(x), and h'(x) is the derivative of h(x).
To apply the quotient rule, follow these steps:
1. Identify g(x) and h(x) in the given function.
2. Find g'(x) and h'(x) by taking the derivatives of g(x) and h(x) separately.
3. Apply the quotient rule formula to find f'(x) by plugging in the values of g'(x), h(x), g(x), and h'(x) into the formula.
4. Simplify the expression to get the final derivative.
Here’s a step-by-step example to illustrate how to use the quotient rule:
Let’s find the derivative of f(x) = (x^2 + 3x) / (2x – 1).
Step 1:
Identify g(x) = x^2 + 3x and h(x) = 2x – 1.
Step 2:
Find g'(x) and h'(x):
g'(x) = d/dx (x^2 + 3x) = 2x + 3 (using the power rule and the constant rule)
h'(x) = d/dx (2x – 1) = 2 (using the constant rule)
Step 3:
Apply the quotient rule formula:
f'(x) = [(2x + 3) * (2x – 1) – (x^2 + 3x) * 2] / [(2x – 1)^2]
Step 4:
Simplify the expression:
f'(x) = (4x^2 – 2x + 6x – 3 – 2x^2 – 6x) / (4x^2 – 4x + 1)
= (2x^2 – 2x – 3) / (4x^2 – 4x + 1)
So, the derivative of f(x) = (x^2 + 3x) / (2x – 1) is f'(x) = (2x^2 – 2x – 3) / (4x^2 – 4x + 1).
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