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Find the derivative of the following function.

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

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

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=\ln \left(-2 x^{5}+x^{4}\right)

\newline

Answer:

y^{\prime}=

Identify function: Identify the function to differentiate. $\newline$ The function given is $y = \ln(-2x^5 + x^4)$ . We need to find the derivative of this function with respect to $x$ .
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 $\ln(u)$ and the inner function is $u = -2x^5 + x^4$ .
Differentiate outer function: Differentiate the outer function. $\newline$ The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$ . So, if $y = \ln(u)$ , then $\frac{dy}{du} = \frac{1}{u}$ .
Differentiate inner function: Differentiate the inner function. $\newline$ The inner function is $u = -2x^5 + x^4$ . We need to find $\frac{du}{dx}$ . $\newline$ $\frac{du}{dx} = \frac{d}{dx}(-2x^5) + \frac{d}{dx}(x^4)$ $\newline$ $\frac{du}{dx} = -10x^4 + 4x^3$
Apply chain rule: Apply the chain rule. $\newline$ Now we combine the derivatives from Step $3$ and Step $4$ using the chain rule. $\newline$ $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ $\newline$ $\frac{dy}{dx} = \left(\frac{1}{u}\right) \cdot (-10x^4 + 4x^3)$
Substitute back: Substitute the inner function back into the derivative. $\newline$ We substitute $u = -2x^5 + x^4$ back into the derivative we found in Step $5$ . $\newline$ $\frac{dy}{dx} = \frac{1}{(-2x^5 + x^4)} \times (-10x^4 + 4x^3)$
Simplify expression: Simplify the expression. $\newline$ We can now simplify the expression to find the final derivative. $\newline$ $\frac{dy}{dx} = \frac{-10x^4 + 4x^3}{-2x^5 + x^4}$

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