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Find the derivative of the following function.

y=9^(5x^(3)+4x^(2))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=9^{5 x^{3}+4 x^{2}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=9^{5 x^{3}+4 x^{2}}

\newline

Answer:

y^{\prime}=

Identify Components: Identify the components of the function. $\newline$ The function $y = 9^{5x^{3}+4x^{2}}$ is an exponential function where the base is a constant ( $9$ ) and the exponent is a polynomial ( $5x^{3}+4x^{2}$ ).
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. In this case, the outer function is $9^u$ and the inner function is $u = 5x^{3}+4x^{2}$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $9^u$ with respect to $u$ is $(9^u \cdot \ln(9))$ . We will later substitute $u$ with the inner function.
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The inner function is $u = 5x^{3}+4x^{2}$ . The derivative of $5x^{3}$ with respect to $x$ is $15x^{2}$ , and the derivative of $4x^{2}$ with respect to $x$ is $8x$ . Therefore, the derivative of the inner function is $u' = 15x^{2} + 8x$ .
Apply Chain Rule Multiplication: Apply the chain rule by multiplying the derivatives of the outer and inner functions. $\newline$ Using the chain rule, the derivative of $y$ with respect to $x$ is $y' = (9^{5x^{3}+4x^{2}} \cdot \ln(9)) \cdot (15x^{2} + 8x)$ .
Simplify Derivative: Simplify the expression for the derivative. $\newline$ $y' = (9^{5x^{3}+4x^{2}} \cdot \ln(9)) \cdot (15x^{2} + 8x)$ is already simplified, and there is no further simplification needed.

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