Find the derivative of the following function. y=6^(-9x^(5)) Answer: y^(')=

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Find the derivative of the following function.  y=6^(-9x^(5)) Answer:  y^(')=

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Identify Function & Rule: Identify the function and the type of differentiation rule to apply. We have the function y = 6^(-9x^(5)). To find the derivative, we need to apply the chain rule because the function is a composition of functions: an exponential function with a base other than 'e' and a power function.Rewrite Using Exponential: Rewrite the function using the natural exponential function for easier differentiation. We can rewrite the function as y = e^(ln(6) * (-9x^(5))) because a^b = e^(ln(a) * b) for any positive a and real number b.Apply Chain Rule: Differentiate the function using the chain rule. The derivative of y with respect to x is y' = d/dx [e^(ln(6) * (-9x^(5)))]. Using the chain rule, we get y' = e^(ln(6) * (-9x^(5))) * d/dx [ln(6) * (-9x^(5))].Differentiate Exponent: Differentiate the exponent ln(6) * (-9x^(5)). The derivative of ln(6) * (-9x^(5)) with respect to x is ln(6) * d/dx [-9x^(5)]. Since ln(6) is a constant, it remains as is, and the derivative of -9x^(5) is -45x^(4). So, we have y' = e^(ln(6) * (-9x^(5))) * ln(6) * (-45x^(4)).Simplify Derivative: Simplify the derivative. We can now simplify the expression by substituting back the original base of the exponential: y' = 6^(-9x^(5)) * ln(6) * (-45x^(4)).Check for Errors: Check for any mathematical errors. Review the differentiation steps to ensure that the chain rule was applied correctly and that the simplification steps are accurate.

Find the derivative of the following function. $\newline$ $y=6^{-9 x^{5}}$ $\newline$ Answer: $y^{\prime}=$

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Find the derivative of the following function.\newliney=6−9x5 y=6^{-9 x^{5}} y=6−9x5\newlineAnswer: y′= y^{\prime}= y′=

Find the derivative of the following function. $\newline$ $y=6^{-9 x^{5}}$ $\newline$ Answer: $y^{\prime}=$