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

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

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Identify Function & Differentiation Type: Identify the function and the type of differentiation required. We are given the function y=4^(9x^(4)) and we need to find its derivative with respect to x. This is an example of a composite function where we have an exponential function with a base of 4 and an exponent that is itself a function of x. To differentiate this, we will use the chain rule.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 4^u where u=9x^(4), and the inner function is u=9x^(4). We will first differentiate the outer function with respect to u, and then multiply it by the derivative of the inner function with respect to x.Differentiate Outer Function: Differentiate the outer function with respect to u. The derivative of 4^u with respect to u is (4^u * ln(4)). This is because the derivative of a^u with respect to u is a^u * ln(a), where ln denotes the natural logarithm.Differentiate Inner Function: Differentiate the inner function with respect to x. The derivative of 9x^(4) with respect to x is 36x^(3). This is because the derivative of x^n with respect to x is n*x^(n-1).Combine Derivatives: Combine the derivatives using the chain rule. Now we multiply the derivative of the outer function by the derivative of the inner function to get the overall derivative of y with respect to x. y' = (4^(9x^(4)) * ln(4)) * 36x^(3)Simplify Expression: Simplify the expression if possible. In this case, there is no further simplification that can be done without changing the form of the expression. So, the derivative of the function y=4^(9x^(4)) with respect to x is y' = (4^(9x^(4)) * ln(4)) * 36x^(3).

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

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