Product Rule
The product rule is a rule in calculus that allows us to find the derivative of the product of two functions
The product rule is a rule in calculus that allows us to find the derivative of the product of two functions. It states that if you have two functions, say f(x) and g(x), then the derivative of their product, denoted as (f(x)g(x))’, is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function.
Mathematically, the product rule can be expressed as:
(f(x)g(x))’ = f'(x)g(x) + f(x)g'(x)
Let’s take a look at an example to understand this concept better.
Example:
Find the derivative of the function f(x) = x^2 * sin(x)
Solution:
To find the derivative of this function using the product rule, we need to differentiate both the x^2 and sin(x) terms separately.
Let’s assign f(x) = x^2 and g(x) = sin(x).
Now, we differentiate f(x) and g(x) individually:
f'(x) = 2x (derivative of x^2 with respect to x)
g'(x) = cos(x) (derivative of sin(x) with respect to x)
Next, we apply the product rule formula:
(f(x)g(x))’ = f'(x)g(x) + f(x)g'(x)
Substituting the values, we get:
((x^2)(sin(x)))’ = (2x)(sin(x)) + (x^2)(cos(x))
Therefore, the derivative of f(x) = x^2 * sin(x) is:
f'(x) = 2x * sin(x) + x^2 * cos(x)
That’s it! We have successfully used the product rule to find the derivative of the given function.
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