Question
Find derivative ofx² sinx
Answers
GPT-4o mini
To find the derivative of the function \( f(x) = x^2 \sin x \), we can use the product rule. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:
\[
(uv)' = u'v + uv'
\]
For our function, let:
- \( u = x^2 \) and \( v = \sin x \)
Now we need to find the derivatives of \( u \) and \( v \):
- \( u' = 2x \)
- \( v' = \cos x \)
Now we apply the product rule:
\[
f'(x) = u'v + uv' = (2x)(\sin x) + (x^2)(\cos x)
\]
So the derivative of \( f(x) = x^2 \sin x \) is:
\[
f'(x) = 2x \sin x + x^2 \cos x
\]
This is the final answer.
\[
(uv)' = u'v + uv'
\]
For our function, let:
- \( u = x^2 \) and \( v = \sin x \)
Now we need to find the derivatives of \( u \) and \( v \):
- \( u' = 2x \)
- \( v' = \cos x \)
Now we apply the product rule:
\[
f'(x) = u'v + uv' = (2x)(\sin x) + (x^2)(\cos x)
\]
So the derivative of \( f(x) = x^2 \sin x \) is:
\[
f'(x) = 2x \sin x + x^2 \cos x
\]
This is the final answer.