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].
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].
