Find the derivative of the following function. y=log_(9)(-5x^(4)-x^(3)) Answer: y^(')=

Identify function and base: Identify the function and the base of the logarithm. We are given the function y = log_9(-5x^4 - x^3). The base of the logarithm is 9.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_9(u) and the inner function is u = -5x^4 - x^3.Differentiate outer function: Differentiate the outer function with respect to the inner function. The derivative of log_9(u) with respect to u is 1/(u * ln(9)), where ln is the natural logarithm.Differentiate inner function: Differentiate the inner function with respect to x. The inner function is u = -5x^4 - x^3. Using the power rule, the derivative is du/dx = -20x^3 - 3x^2.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/(-5x^4 - x^3 * ln(9))) * (-20x^3 - 3x^2).Simplify expression: Simplify the expression. We can factor out x^2 from the numerator and combine the terms to get the final derivative. dy/dx = (-20x^3 - 3x^2) / ((-5x^4 - x^3) * ln(9)) dy/dx = x^2(-20x - 3) / (x^3(-5x - 1) * ln(9)) dy/dx = (-20x - 3) / (x(-5x - 1) * ln(9))

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Find the derivative of the following function.

y=log_(9)(-5x^(4)-x^(3))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\log _{9}\left(-5 x^{4}-x^{3}\right)$ $\newline$ Answer: $y^{\prime}=$

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Precalculus

Quotient property of logarithms

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Q. Find the derivative of the following function.

\newline

y=\log _{9}\left(-5 x^{4}-x^{3}\right)

\newline

Answer:

y^{\prime}=

Identify function and base: Identify the function and the base of the logarithm. $\newline$ We are given the function $y = \log_9(-5x^4 - x^3)$ . The base of the logarithm is $9$ .
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 $\log_9(u)$ and the inner function is $u = -5x^4 - x^3$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $\log_9(u)$ with respect to $u$ is $\frac{1}{u \cdot \ln(9)}$ , where $\ln$ is the natural logarithm.
Differentiate inner function: Differentiate the inner function with respect to $x$ . The inner function is $u = -5x^4 - x^3$ . Using the power rule, the derivative is $\frac{du}{dx} = -20x^3 - 3x^2$ .
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 $\frac{dy}{dx} = \frac{1}{(-5x^4 - x^3 \cdot \ln(9))} \cdot (-20x^3 - 3x^2)$ .
Simplify expression: Simplify the expression. $\newline$ We can factor out $x^2$ from the numerator and combine the terms to get the final derivative. $\newline$ $\frac{dy}{dx} = \frac{-20x^3 - 3x^2}{(-5x^4 - x^3) \cdot \ln(9)}$ $\newline$ $\frac{dy}{dx} = \frac{x^2(-20x - 3)}{x^3(-5x - 1) \cdot \ln(9)}$ $\newline$ $\frac{dy}{dx} = \frac{-20x - 3}{x(-5x - 1) \cdot \ln(9)}$