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Find the derivative of the following function.

y=6^(-9x^(5))
Answer:
y^(')=

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

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Find derivatives using logarithmic differentiation

Full solution

Q. Find the derivative of the following function.

\newline

y=6^{-9 x^{5}}

\newline

Answer:

y^{\prime}=

Identify Function & Rule: Identify the function and the type of differentiation rule to apply. $\newline$ 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. $\newline$ 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. $\newline$ The derivative of $y$ with respect to $x$ is $y' = \frac{d}{dx} [e^{(\ln(6) * (-9x^{5}))}]$ . $\newline$ Using the chain rule, we get $y' = e^{(\ln(6) * (-9x^{5}))} * \frac{d}{dx} [\ln(6) * (-9x^{5})]$ .
Differentiate Exponent: Differentiate the exponent $\ln(6) \cdot (-9x^{5})$ . The derivative of $\ln(6) \cdot (-9x^{5})$ with respect to $x$ is $\ln(6) \cdot \frac{d}{dx} [-9x^{5}]$ . Since $\ln(6)$ is a constant, it remains as is, and the derivative of $-9x^{5}$ is $-45x^{4}$ . So, we have $y' = e^{\ln(6) \cdot (-9x^{5})} \cdot \ln(6) \cdot (-45x^{4})$ .
Simplify Derivative: Simplify the derivative. $\newline$ We can now simplify the expression by substituting back the original base of the exponential: $\newline$ $y' = 6^{-9x^{5}} \cdot \ln(6) \cdot (-45x^{4})$ .
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.

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