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$16^{x^2}+y+16^x+y^2$

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Find derivatives using the product rule II

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Q.

16^{x^2}+y+16^x+y^2

Identify terms and apply derivatives: Identify the terms involving $x$ and apply the derivative rules separately to each term. $\newline$ - Derivative of $16^{x^2}$ with respect to $x$ : Using the chain rule, $\frac{d}{dx}[16^{x^2}] = 16^{x^2} \cdot \ln(16) \cdot 2x$ . $\newline$ - Derivative of $y$ with respect to $x$ : Since $y$ is treated as a constant with respect to $x$ , $\frac{dy}{dx} = 0$ . $\newline$ - Derivative of $16^x$ with respect to $x$ : Using the exponential rule, $16^{x^2}$ $1$ . $\newline$ - Derivative of $16^{x^2}$ $2$ with respect to $x$ : Since $16^{x^2}$ $2$ is treated as a constant with respect to $x$ , $16^{x^2}$ $6$ .
Combine derivatives for entire function: Combine the derivatives to form the derivative of the entire function. $\newline$ - $f'(x) = 16^{x^2} \cdot \ln(16) \cdot 2x + 0 + 16^x \cdot \ln(16) + 0$ . $\newline$ - Simplify the expression: $f'(x) = 16^{x^2} \cdot 2x \cdot \ln(16) + 16^x \cdot \ln(16)$ .

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