Find the derivative of the following function. y=3^(x^(3)) Answer: y^(')=

Identify Function: Identify the function to differentiate. 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. 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. The derivative of 3^u with respect to u is 3^u * 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. y' = (3^u * ln(3)) * (3x^2) Since u = x^3, we substitute back to get: y' = (3^(x^3) * ln(3)) * (3x^2)Simplify Derivative: Simplify the expression for the derivative. y' = 3^(x^3) * ln(3) * 3x^2 y' = 3x^2 * ln(3) * 3^(x^3)

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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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Calculus

Find derivatives using logarithmic differentiation

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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. $\newline$ $y' = 3^{x^3} \cdot \ln(3) \cdot 3x^2$ $\newline$ $y' = 3x^2 \cdot \ln(3) \cdot 3^{x^3}$