Find the derivative of the following function. y=9^(5x^(3)+4x^(2)) Answer: y^(')=

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Find the derivative of the following function.  y=9^(5x^(3)+4x^(2)) Answer:  y^(')=

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Identify Components: Identify the components of the function. 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. 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. The derivative of 9^u with respect to u is (9^u * 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. Using the chain rule, the derivative of y with respect to x is y' = (9^(5x^(3)+4x^(2)) * ln(9)) * (15x^(2) + 8x).Simplify Derivative: Simplify the expression for the derivative. y' = (9^(5x^(3)+4x^(2)) * ln(9)) * (15x^(2) + 8x) is already simplified, and there is no further simplification needed.

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