Answer :
To find the value of [tex]\( g'(3) \)[/tex] for the function [tex]\( g(x) = x^2 f(x) \)[/tex] where [tex]\( f(3) = 4 \)[/tex] and [tex]\( f'(3) = 5 \)[/tex], we can use the product rule of differentiation. Here's a step-by-step solution:
1. Identify Functions:
- Let [tex]\( u(x) = x^2 \)[/tex].
- Let [tex]\( v(x) = f(x) \)[/tex].
2. Differentiate Each Function:
- The derivative of [tex]\( u(x) = x^2 \)[/tex] is [tex]\( u'(x) = 2x \)[/tex].
- The derivative of [tex]\( v(x) = f(x) \)[/tex] is [tex]\( v'(x) = f'(x) \)[/tex].
3. Apply the Product Rule:
- The product rule states that the derivative of a product [tex]\( uv \)[/tex] is [tex]\( (uv)' = u'v + uv' \)[/tex].
- Applying this to [tex]\( g(x) = x^2 f(x) \)[/tex], we get:
[tex]\[
g'(x) = (x^2)' f(x) + x^2 f'(x) = 2x \cdot f(x) + x^2 \cdot f'(x)
\][/tex]
4. Substitute [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 4 \)[/tex], and [tex]\( f'(3) = 5 \)[/tex] into the Product Rule:
- Substitute into the derivative:
[tex]\[
g'(3) = 2 \cdot 3 \cdot f(3) + 3^2 \cdot f'(3)
\][/tex]
- Plug in the given values:
[tex]\[
g'(3) = 2 \cdot 3 \cdot 4 + 3^2 \cdot 5
\][/tex]
5. Calculate [tex]\( g'(3) \)[/tex]:
- Compute the first part: [tex]\( 2 \times 3 \times 4 = 24 \)[/tex].
- Compute the second part: [tex]\( 9 \times 5 = 45 \)[/tex].
- Add them together: [tex]\( 24 + 45 = 69 \)[/tex].
The value of [tex]\( g'(3) \)[/tex] is [tex]\(\boxed{69}\)[/tex].
1. Identify Functions:
- Let [tex]\( u(x) = x^2 \)[/tex].
- Let [tex]\( v(x) = f(x) \)[/tex].
2. Differentiate Each Function:
- The derivative of [tex]\( u(x) = x^2 \)[/tex] is [tex]\( u'(x) = 2x \)[/tex].
- The derivative of [tex]\( v(x) = f(x) \)[/tex] is [tex]\( v'(x) = f'(x) \)[/tex].
3. Apply the Product Rule:
- The product rule states that the derivative of a product [tex]\( uv \)[/tex] is [tex]\( (uv)' = u'v + uv' \)[/tex].
- Applying this to [tex]\( g(x) = x^2 f(x) \)[/tex], we get:
[tex]\[
g'(x) = (x^2)' f(x) + x^2 f'(x) = 2x \cdot f(x) + x^2 \cdot f'(x)
\][/tex]
4. Substitute [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 4 \)[/tex], and [tex]\( f'(3) = 5 \)[/tex] into the Product Rule:
- Substitute into the derivative:
[tex]\[
g'(3) = 2 \cdot 3 \cdot f(3) + 3^2 \cdot f'(3)
\][/tex]
- Plug in the given values:
[tex]\[
g'(3) = 2 \cdot 3 \cdot 4 + 3^2 \cdot 5
\][/tex]
5. Calculate [tex]\( g'(3) \)[/tex]:
- Compute the first part: [tex]\( 2 \times 3 \times 4 = 24 \)[/tex].
- Compute the second part: [tex]\( 9 \times 5 = 45 \)[/tex].
- Add them together: [tex]\( 24 + 45 = 69 \)[/tex].
The value of [tex]\( g'(3) \)[/tex] is [tex]\(\boxed{69}\)[/tex].
