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Let
H(x)=3x+int_(1)^(x^(2))g(x)dx
(c) Find
H^(')(2) and
H^('')(2).

Let $H(x)=3 x+\int_{1}^{x^{2}} g(x) d x$ $\newline$ (c) Find $H^{\prime}(2)$ and $H^{\prime \prime}(2)$ .

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Evaluate definite integrals using the chain rule

Full solution

Q. Let

H(x)=3 x+\int_{1}^{x^{2}} g(x) d x

\newline

(c) Find

H^{\prime}(2)

and

H^{\prime \prime}(2)

.

Calculate $H'(x)$ : To find $H'(x)$ , we use the Fundamental Theorem of Calculus for the integral part and differentiate the rest normally. $\newline$ $H'(x) = \frac{d}{dx} [3x] + \frac{d}{dx} \left[\int_{1}^{x^2} g(t) \, dt\right]$ $\newline$ $H'(x) = 3 + 2x \cdot g(x^2)$
Find $H'(2)$ : Now we plug in $x=2$ to find $H'(2)$ . $\newline$ $H'(2) = 3 + 2 \times 2 \times g(2^2)$ $\newline$ $H'(2) = 3 + 4 \times g(4)$
Differentiate $H''(x)$ : To find $H''(x)$ , we differentiate $H'(x)$ .
$H''(x) = \frac{d}{dx} [3 + 2x \cdot g(x^2)]$
$H''(x) = 0 + 2 \cdot g(x^2) + 2x \cdot \frac{d}{dx} [g(x^2)]$
$H''(x) = 2 \cdot g(x^2) + 4x \cdot g'(x^2)$
Find $H''(2)$ : Now we plug in $x=2$ to find $H''(2)$ .
$H''(2) = 2 \times g(4) + 4\times2 \times g'(4)$
$H''(2) = 2 \times g(4) + 8 \times g'(4)$

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