d/dx[f(x)g(x)] (product rule)
To find the derivative of the product of two functions, f(x) and g(x), we use the product rule
To find the derivative of the product of two functions, f(x) and g(x), we use the product rule. The formula for the derivative of the product of two functions is:
d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Here, f'(x) represents the derivative of f(x) with respect to x and g'(x) represents the derivative of g(x) with respect to x.
Let’s use the product rule to find the derivative of f(x)g(x).
Step 1: Find the derivative of f(x), denoted as f'(x).
Step 2: Find the derivative of g(x), denoted as g'(x).
Step 3: Use the product rule formula to find the derivative of f(x)g(x).
Let’s go through an example for better understanding:
Example: Find the derivative of f(x) = 3x^2 * sin(x)
Step 1: Find f'(x)
To find f'(x), we need to differentiate f(x) = 3x^2 * sin(x) with respect to x.
f'(x) = 2 * 3x^(2-1) * sin(x) + 3x^2 * cos(x)
= 6x * sin(x) + 3x^2 * cos(x)
Step 2: Find g'(x)
In this example, g(x) is sin(x). The derivative of sin(x) is cos(x).
g'(x) = cos(x)
Step 3: Use the product rule formula to find d/dx[f(x)g(x)]
According to the product rule formula:
d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Substituting the values we found earlier:
d/dx[(3x^2 * sin(x))] = (6x * sin(x) + 3x^2 * cos(x)) * sin(x) + (3x^2 * sin(x)) * cos(x)
Simplifying the expression:
d/dx[(3x^2 * sin(x))] = 6x * sin^2(x) + 3x^2 * sin(x) * cos(x) + 3x^2 * sin(x) * cos(x)
Therefore, the derivative of f(x) = 3x^2 * sin(x) is:
d/dx[3x^2 * sin(x)] = 6x * sin^2(x) + 6x^2 * sin(x) * cos(x)
So, the answer is 6x * sin^2(x) + 6x^2 * sin(x) * cos(x).
More Answers:
[next_post_link]