Find the derivative of the following function. y=log_(2)(-6x^(2)-6x) Answer: y^(')=

Identify function and base: Identify the function and the base of the logarithm. We are given the function y = log_2(-6x^2 - 6x), where the base of the logarithm is 2.Apply chain rule: Apply the chain rule for differentiation. 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 log_2(u) and the inner function is u = -6x^2 - 6x.Differentiate outer function: Differentiate the outer function with respect to the inner function. The derivative of log_2(u) with respect to u is 1/(u * ln(2)), where ln is the natural logarithm.Differentiate inner function: Differentiate the inner function with respect to x. The inner function is u = -6x^2 - 6x. Its derivative with respect to x is du/dx = -12x - 6.Apply chain rule multiplication: Apply the chain rule by multiplying the derivatives from Step 3 and Step 4. The derivative of y with respect to x is dy/dx = (1/((-6x^2 - 6x) * ln(2))) * (-12x - 6).Simplify expression: Simplify the expression. We can simplify the derivative to get y' = (-12x - 6) / ((-6x^2 - 6x) * ln(2)).

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Find the derivative of the following function.

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

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

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Precalculus

Quotient property of logarithms

Full solution

Q. Find the derivative of the following function.

\newline

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

\newline

Answer:

y^{\prime}=

Identify function and base: Identify the function and the base of the logarithm. $\newline$ We are given the function $y = \log_2(-6x^2 - 6x)$ , where the base of the logarithm is $2$ .
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. $\newline$ In this case, the outer function is $\log_2(u)$ and the inner function is $u = -6x^2 - 6x$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $\log_2(u)$ with respect to $u$ is $\frac{1}{u \cdot \ln(2)}$ , where $\ln$ is the natural logarithm.
Differentiate inner function: Differentiate the inner function with respect to $x$ . The inner function is $u = -6x^2 - 6x$ . Its derivative with respect to $x$ is $\frac{du}{dx} = -12x - 6$ .
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 $\frac{dy}{dx} = \frac{1}{((-6x^2 - 6x) \cdot \ln(2))} \cdot (-12x - 6)$ .
Simplify expression: Simplify the expression. $\newline$ We can simplify the derivative to get $y' = \frac{-12x - 6}{(-6x^2 - 6x) \cdot \ln(2)}$ .