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Find the derivative of the following function.

y=8^(-9x^(5)+6x^(4))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=8^{-9 x^{5}+6 x^{4}}$ $\newline$ 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}+6 x^{4}}

\newline

Answer:

y^{\prime}=

Identify Function Components: Identify the function and its components. $\newline$ We have $y = 8^{(-9x^{5} + 6x^{4})}$ . This is an exponential function with a base of $8$ and an exponent of $-9x^5 + 6x^4$ .
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 $u(x) = -9x^5 + 6x^4$ .
Derivative of Outer Function: Find the derivative of the outer function $a(u)$ with respect to $u$ . The derivative of $a^u$ with respect to $u$ is $a^u \cdot \ln(a)$ , where $a$ is a constant. So, $a'(u) = 8^u \cdot \ln(8)$ .
Derivative of Inner Function: Find the derivative of the inner function $u(x)$ with respect to $x$ . The inner function is $u(x) = -9x^5 + 6x^4$ . Using the power rule, the derivative is $u'(x) = \frac{d}{dx}(-9x^5) + \frac{d}{dx}(6x^4) = -45x^4 + 24x^3$ .
Apply Chain Rule with Derivatives: Apply the chain rule using the derivatives from steps $3$ and $4$ . $\newline$ The chain rule gives us $y'(x) = a'(u(x)) \cdot u'(x)$ . $\newline$ Substitute $a'(u)$ and $u'(x)$ into the equation. $\newline$ $y'(x) = (8^{-9x^5 + 6x^4} \cdot \ln(8)) \cdot (-45x^4 + 24x^3)$ .
Simplify Derivative: Simplify the expression for the derivative. $\newline$ $y'(x) = 8^{(-9x^5 + 6x^4)} \cdot \ln(8) \cdot (-45x^4 + 24x^3)$ . $\newline$ This is the simplified form of the derivative.

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