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Find the derivative of the following function.

y=3^(x^(3))
Answer:
y^(')=

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

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Find derivatives using logarithmic differentiation

Full solution

Q. Find the derivative of the following function.

\newline

y=3^{x^{3}}

\newline

Answer:

y^{\prime}=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = 3^{x^3}$ . We need to find its derivative with respect to $x$ , which is denoted as $y'$ .
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. In this case, the outer function is $3^{u}$ (where $u = x^{3}$ ) and the inner function is $u = x^{3}$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $3^u$ with respect to $u$ is $3^u \cdot \ln(3)$ , where $\ln(3)$ is the natural logarithm of $3$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $u = x^3$ with respect to $x$ is $3x^2$ .
Apply Chain Rule: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ $y' = (3^u \cdot \ln(3)) \cdot (3x^2)$ $\newline$ Since $u = x^3$ , we substitute back to get: $\newline$ $y' = (3^{x^3} \cdot \ln(3)) \cdot (3x^2)$
Simplify Derivative: Simplify the expression for the derivative. $\newline$ $y' = 3^{x^3} \cdot \ln(3) \cdot 3x^2$ $\newline$ $y' = 3x^2 \cdot \ln(3) \cdot 3^{x^3}$

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