Product Rule
The product rule is a rule in calculus that allows us to find the derivative of a product of two functions
The product rule is a rule in calculus that allows us to find the derivative of a product of two functions. It is commonly used when differentiating equations involving multiple terms or variables.
The general formula for the product rule is:
(d/dx)(f(x) * g(x)) = f'(x) * g(x) + f(x) * g'(x)
where f(x) and g(x) are two differentiable functions and f'(x) and g'(x) are their respective derivatives.
To better understand the product rule, let’s work through an example.
Example:
Find the derivative of the function h(x) = (2x^2 + 3x)(4x – 1).
Step 1: Identify the functions f(x) and g(x).
In this case, f(x) = 2x^2 + 3x and g(x) = 4x – 1.
Step 2: Differentiate each function.
Find the derivatives of f(x) and g(x) using the power rule and the constant rule, respectively:
f'(x) = 4x + 3
g'(x) = 4
Step 3: Apply the product rule formula.
Using the product rule formula:
h'(x) = f'(x) * g(x) + f(x) * g'(x)
h'(x) = (4x + 3) * (4x – 1) + (2x^2 + 3x) * 4
Step 4: Simplify the expression.
Distribute the terms and combine like terms:
h'(x) = 16x^2 – 4x + 12x – 3 + 8x^2 + 12x
= 24x^2 + 20x – 3
So, the derivative of the function h(x) = (2x^2 + 3x)(4x – 1) is h'(x) = 24x^2 + 20x – 3.
Remember, the product rule is a helpful tool when differentiating functions that involve multiplication. By following the steps mentioned above, you can find the derivative of any product function.
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