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In this class, we will learn about Product Rule of Differentiation, its formula, and some examples that demonstrate the use of product rule.
What is the Product Rule?
The product rule is a derivative rule used to find the derivative of the product of two functions.
Product Rule states that if you have a function that is the product of two other functions, the derivative of this function is given by:
(f(x) * g(x))' == f'(x) * g(x) + f(x) * g'(x)
where f(x) and g(x) are differentiable functions of x.
Examples for Product Rule
1. Find the derivative of the function h(x) == x^2 * ln(x).
Solution
Given: h(x) == x^2 * ln(x)
h(x) is of the form f(x)*g(x), where
f(x) == x^2
g(x) == ln(x)
Find: We have to find the derivative of h(x).
Applying Formula: Since h(x) is in the form of product of two functions, let us apply product rule.
(f(x) * g(x))' == f'(x) * g(x) + f(x) * g'(x)
h'(x) == (f(x) * g(x))'
h'(x) == f'(x) * g(x) + f(x) * g'(x) ―――①
f(x) == x^2 ⇒ f'(x) == 2x
g(x) == ln(x) ⇒ g'(x) == 1/x
Substitute the values of f(x), g(x), f'(x), and g'(x) in ①
h'(x) == f'(x) * g(x) + f(x) * g'(x)
h'(x) == (2x) * ln(x) + (x^2) * (1/x)
h'(x) == 2x*ln(x) + x
Therefore,
h'(x) == 2x*ln(x) + x
2. Find the derivative of the function h(x) == sin(x) * cos(x).
Solution
Given: h(x) == sin(x) * cos(x)
h(x) is of the form f(x)*g(x), where
f(x) == sin(x)
g(x) == cos(x)
Find: We have to find the derivative of h(x).
Applying Formula: Since h(x) is in the form of product of two functions, let us apply product rule.
(f(x) * g(x))' == f'(x) * g(x) + f(x) * g'(x)
h'(x) == (f(x) * g(x))'
h'(x) == f'(x) * g(x) + f(x) * g'(x) ―――①
f(x) == sin(x) ⇒ f'(x) == cos(x)
g(x) == cos(x) ⇒ g'(x) == -sin(x)
Substitute the values of f(x), g(x), f'(x), and g'(x) in ①
h'(x) == f'(x) * g(x) + f(x) * g'(x)
h'(x) == cos(x) * cos(x) + sin(x) * (-sin(x))
h'(x) == cos^2 (x) - sin^2 (x)
Therefore,
h'(x) == cos^2 (x) - sin^2 (x)