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Find the derivative of the following function.

y=2^(-8x^(5)+2x^(4))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=2^{-8 x^{5}+2 x^{4}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=2^{-8 x^{5}+2 x^{4}}

\newline

Answer:

y^{\prime}=

Identify Function Components: Identify the function and its components. $\newline$ We have the function $y = 2^{-8x^{5} + 2x^{4}}$ . This is an exponential function with a base of $2$ and an exponent that is a polynomial in $x$ .
Apply Chain Rule: Apply the chain rule for differentiation. $\newline$ 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. In this case, the outer function is $2^u$ and the inner function is $u = -8x^{5} + 2x^{4}$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function $u$ . The derivative of $2^u$ with respect to $u$ is $(2^u) \cdot \ln(2)$ , where $\ln(2)$ is the natural logarithm of $2$ .
Differentiate Inner Function: Differentiate the inner function $u = -8x^{5} + 2x^{4}$ with respect to $x$ . The derivative of $-8x^{5}$ is $-40x^{4}$ , and the derivative of $2x^{4}$ is $8x^{3}$ . So, the derivative of the inner function $u$ with respect to $x$ is $u' = -40x^{4} + 8x^{3}$ .
Apply Chain Rule Multiplication: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ The derivative of $y$ with respect to $x$ is $y' = (2^{-8x^{5} + 2x^{4}}) \cdot \ln(2) \cdot (-40x^{4} + 8x^{3})$ .
Simplify Derivative Expression: Simplify the expression for the derivative. $\newline$ $y' = (2^{-8x^{5} + 2x^{4}}) \cdot \ln(2) \cdot (-40x^{4} + 8x^{3})$ can be left as is, or factored to show the common factor of $8x^{3}$ . $\newline$ $y' = 8x^{3} \cdot (2^{-8x^{5} + 2x^{4}}) \cdot \ln(2) \cdot (-5x + 1)$

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