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Find the derivative of the following function.

y=3^(-8x^(2))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=3^{-8 x^{2}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=3^{-8 x^{2}}

\newline

Answer:

y^{\prime}=

Identify Function Components: Identify the function and its components. $\newline$ We have $y = 3^{-8x^{2}}$ . This is an exponential function where the base is a constant ( $3$ ) and the exponent is a function of $x$ ( $-8x^{2}$ ).
Apply Chain Rule: Apply the chain rule for differentiation. $\newline$ The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. $\newline$ In this case, the outer function is $f(u) = 3^u$ and the inner function is $u(x) = -8x^{2}$ .
Differentiate Outer Function: Differentiate the outer function with respect to $u$ . The derivative of $f(u) = 3^u$ with respect to $u$ is $f'(u) = 3^u \cdot \ln(3)$ , where $\ln(3)$ is the natural logarithm of $3$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $u(x) = -8x^{2}$ with respect to $x$ is $u'(x) = \frac{d}{dx}(-8x^{2}) = -16x$ .
Apply Chain Rule with Derivatives: Apply the chain rule using the derivatives from steps $3$ and $4$ . $\newline$ $y' = f'(u(x)) \cdot u'(x) = (3^{-8x^{2}} \cdot \ln(3)) \cdot (-16x)$ .
Simplify Derivative: Simplify the expression for the derivative. $\newline$ $y' = -16x \cdot 3^{-8x^{2}} \cdot \ln(3)$ . $\newline$ This is the derivative of the function $y$ with respect to $x$ .

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