Let f be a function such that f(2)=3 and f^(')(2)=-1. Let g be the function g(x)=x^(2). Let G be a function defined as G(x)=f(x)*g(x). G^(')(2)=

Identify Product Rule: G(x) = f(x) * g(x), so we need to use the product rule to find G'(x). The product rule is (f * g)' = f' * g + f * g'.Find g'(x): First, find g'(x) since g(x) = x^2. g'(x) = 2x.Evaluate g'(2): Now, evaluate g'(2) by substituting x with 2. g'(2) = 2 * 2 = 4.Determine f(2) and f'(2): We already know f(2) = 3 and f'(2) = -1 from the problem statement.Apply Product Rule Formula: Now, apply the product rule: G'(x) = f'(x) * g(x) + f(x) * g'(x).Substitute x with 2: Substitute x with 2 to find G'(2): G'(2) = f'(2) * g(2) + f(2) * g'(2).Calculate G'(2): Calculate G'(2) using the known values: G'(2) = (-1) * (2^2) + 3 * 4.Simplify Expression: Simplify the expression: G'(2) = (-1) * 4 + 3 * 4.Final Result: G'(2) = -4 + 12.G'(2) = 8.

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Let
f be a function such that
f(2)=3 and
f^(')(2)=-1.
Let
g be the function
g(x)=x^(2).

Let
G be a function defined as
G(x)=f(x)*g(x).

G^(')(2)=

- Let $f$ be a function such that $f(2)=3$ and $f^{\prime}(2)=-1$ . $\newline$ - Let $g$ be the function $g(x)=x^{2}$ . $\newline$ Let $G$ be a function defined as $G(x)=f(x) \cdot g(x)$ . $\newline$ $G^{\prime}(2)=$

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Math Problems

Algebra 2

Evaluate exponential functions

Full solution

Q. - Let

f

be a function such that

f(2)=3

and

f^{\prime}(2)=-1

\newline

- Let

g

be the function

g(x)=x^{2}

\newline

Let

G

be a function defined as

G(x)=f(x) \cdot g(x)

\newline

G^{\prime}(2)=

Identify Product Rule: $G(x) = f(x) \cdot g(x)$ , so we need to use the product rule to find $G'(x)$ . The product rule is $(f \cdot g)' = f' \cdot g + f \cdot g'$ .
Find $g'(x)$ : First, find $g'(x)$ since $g(x) = x^2$ . $\newline$ $g'(x) = 2x$ .
Evaluate $g'(2)$ : Now, evaluate $g'(2)$ by substituting $x$ with $2$ . $\newline$ $g'(2) = 2 \times 2 = 4$ .
Determine $f(2)$ and $f'(2)$ : We already know $f(2) = 3$ and $f'(2) = -1$ from the problem statement.
Apply Product Rule Formula: Now, apply the product rule: $G'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$ .
Substitute $x$ with $2$ : Substitute $x$ with $2$ to find $G'(2)$ : $G'(2) = f'(2) \cdot g(2) + f(2) \cdot g'(2)$ .
Calculate $G'(2)$ : Calculate $G'(2)$ using the known values: $G'(2) = (-1) \times (2^2) + 3 \times 4$ .
Simplify Expression: Simplify the expression: $G'(2) = (-1) \times 4 + 3 \times 4$ .
Final Result: $G'(2) = -4 + 12$ .
Final Result: $G'(2) = -4 + 12$ . $G'(2) = 8$ .