College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Let [tex]\( f \)[/tex] be a differentiable function such that [tex]\( f(3) = 4 \)[/tex] and [tex]\( f^{\prime}(3) = 5 \)[/tex].

If [tex]\( g(x) = x^2 f(x) \)[/tex], what is the value of [tex]\( g^{\prime}(3) \)[/tex]?

A. 11
B. 24
C. 30
D. 69

Answer :

To find the value of [tex]\( g'(3) \)[/tex], we need to differentiate the function [tex]\( g(x) = x^2 f(x) \)[/tex] using the product rule.

The product rule states that if you have two functions multiplied together, such as [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex], then the derivative is given by:
[tex]\[
(uv)' = u'v + uv'
\][/tex]

For [tex]\( g(x) = x^2 f(x) \)[/tex], we identify:
- [tex]\( u(x) = x^2 \)[/tex] and thus [tex]\( u'(x) = 2x \)[/tex]
- [tex]\( v(x) = f(x) \)[/tex] and thus [tex]\( v'(x) = f'(x) \)[/tex]

Now, applying the product rule:
[tex]\[
g'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\][/tex]

Substituting the derivatives we identified:
[tex]\[
g'(x) = (2x) \cdot f(x) + (x^2) \cdot f'(x)
\][/tex]

Now, plug in the values at [tex]\( x = 3 \)[/tex] since we want [tex]\( g'(3) \)[/tex]:
- [tex]\( f(3) = 4 \)[/tex]
- [tex]\( f'(3) = 5 \)[/tex]

Thus the expression becomes:
[tex]\[
g'(3) = (2 \cdot 3) \cdot f(3) + (3^2) \cdot f'(3)
\][/tex]

Calculate each term:
1. [tex]\( (2 \cdot 3) \cdot f(3) = 6 \cdot 4 = 24 \)[/tex]
2. [tex]\( (3^2) \cdot f'(3) = 9 \cdot 5 = 45 \)[/tex]

Add these results:
[tex]\[
g'(3) = 24 + 45 = 69
\][/tex]

Therefore, the value of [tex]\( g^{\prime}(3) \)[/tex] is [tex]\(\boxed{69}\)[/tex].