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Find the derivative of the following function.

y=log_(6)(-2x^(4)-7x^(3))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\log _{6}\left(-2 x^{4}-7 x^{3}\right)$ $\newline$ Answer: $y^{\prime}=$

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Quotient property of logarithms

Full solution

Q. Find the derivative of the following function.

\newline

y=\log _{6}\left(-2 x^{4}-7 x^{3}\right)

\newline

Answer:

y^{\prime}=

Identify Function & Derivative Type: Identify the function and the type of derivative to be found. $\newline$ We need to find the derivative of the function $y$ with respect to $x$ , where $y = \log_6(-2x^4 - 7x^3)$ . This is a logarithmic differentiation problem.
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. For the logarithmic function $y = \log_6(u)$ , where $u = -2x^4 - 7x^3$ , the derivative with respect to $x$ is $y' = (1 / (u \cdot \ln(6))) \cdot \frac{du}{dx}$ .
Differentiate Inner Function: Differentiate the inner function $u = -2x^4 - 7x^3$ with respect to $x$ . The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = \frac{d}{dx}(-2x^4) + \frac{d}{dx}(-7x^3) = -8x^3 - 21x^2$ .
Substitute Derivative into Formula: Substitute the derivative of the inner function into the chain rule formula. $\newline$ Now we have $y' = \frac{1}{((-2x^4 - 7x^3) \cdot \ln(6))} \cdot (-8x^3 - 21x^2)$ .
Simplify Derivative: Simplify the expression for the derivative. $\newline$ We can leave the derivative in its current form, as it is fully simplified and each term is in terms of $x$ .

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