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Find the derivative $\newline$ $2(x^{2}+4)$

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Find derivatives of exponential functions

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Q. Find the derivative

\newline

2(x^{2}+4)

Apply rules for differentiation: Apply the constant multiple rule and the sum rule for differentiation. $\newline$ The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function. The sum rule states that the derivative of a sum of functions is the sum of their derivatives. $\newline$ So, we will differentiate the function $2(x^{2}+4)$ by applying these rules. $\newline$ $f(x) = 2(x^{2}+4)$ $\newline$ $f'(x) = 2 \cdot \frac{d}{dx}(x^{2}) + 2 \cdot \frac{d}{dx}(4)$
Differentiate individual terms: Differentiate the individual terms. $\newline$ The derivative of $x^{2}$ with respect to $x$ is $2x$ , using the power rule which states that $\frac{d}{dx}(x^n) = n\cdot x^{(n-1)}$ . $\newline$ The derivative of a constant, like $4$ , is $0$ . $\newline$ So, we have: $\newline$ $f'(x) = 2 \cdot (2x) + 2 \cdot 0$
Simplify the expression: Simplify the expression. $\newline$ Now we simplify the expression by multiplying the constants and adding up the terms. $\newline$ $f'(x) = 4x + 0$ $\newline$ $f'(x) = 4x$

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