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Find the derivative of the following function.

y=8^(9x^(5)-5x^(4))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=8^{9 x^{5}-5 x^{4}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=8^{9 x^{5}-5 x^{4}}

\newline

Answer:

y^{\prime}=

Identify Function Components: Identify the function and its components. $\newline$ We have $y = 8^{9x^{5} - 5x^{4}}$ . This is an exponential function with a base of $8$ and an exponent of $9x^5 - 5x^4$ .
Apply Chain Rule: Apply the chain rule for derivatives. $\newline$ 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. $\newline$ Here, the outer function is $a(u) = 8^u$ and the inner function is $u(x) = 9x^5 - 5x^4$ .
Derivative of Outer Function: Find the derivative of the outer function $a(u) = 8^u$ with respect to $u$ . The derivative of $a^u$ with respect to $u$ is $a^u \cdot \ln(a)$ , where $a$ is a constant and $u$ is the variable. So, $a'(u) = 8^u \cdot \ln(8)$ .
Derivative of Inner Function: Find the derivative of the inner function $u(x) = 9x^5 - 5x^4$ with respect to $x$ . The derivative of $9x^5$ with respect to $x$ is $45x^4$ , and the derivative of $-5x^4$ with respect to $x$ is $-20x^3$ . So, $u'(x) = 45x^4 - 20x^3$ .
Apply Chain Rule with Derivatives: Apply the chain rule using the derivatives from steps $3$ and $4$ . $\newline$ $y' = a'(u) \cdot u'(x)$ $\newline$ $y' = (8^{9x^5 - 5x^4} \cdot \ln(8)) \cdot (45x^4 - 20x^3)$
Simplify Derivative: Simplify the expression for the derivative. $\newline$ $y' = 8^{9x^5 - 5x^4} \cdot \ln(8) \cdot (45x^4 - 20x^3)$ $\newline$ This is the final form of the derivative.

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