Find the derivative of the following function. $\newline$ $y=\log _{8}\left(-9 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$

View step-by-step help

Home

Math Problems

Precalculus

Quotient property of logarithms

Full solution

Q. Find the derivative of the following function.

\newline

y=\log _{8}\left(-9 x^{5}\right)

\newline

Answer:

y^{\prime}=

Understand and apply chain rule: Understand the function and apply the chain rule. $\newline$ The function $y=\log_{8}(-9x^{5})$ is a logarithmic function with base $8$ . To find the derivative, we need to use the chain rule, which 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.
Convert base to natural logarithm: Convert the base of the logarithm to the natural logarithm ( $\ln$ ). $\newline$ The derivative of a logarithm with base $8$ can be found by converting it to a natural logarithm using the change of base formula: $\log_{a}(b) = \frac{\ln(b)}{\ln(a)}$ . So, $y=\log_{8}(-9x^{5})$ becomes $y=\frac{\ln(-9x^{5})}{\ln(8)}$ .
Differentiate outer function: Differentiate the outer function. $\newline$ The outer function is $\ln(-9x^{5})$ . The derivative of $\ln(u)$ , where $u$ is a function of $x$ , is $\frac{1}{u} \cdot u'$ , where $u'$ is the derivative of $u$ . So, the derivative of $\ln(-9x^{5})$ with respect to $x$ is $\frac{1}{-9x^{5}}$ times the derivative of $\ln(u)$ $0$ with respect to $x$ .
Differentiate inner function: Differentiate the inner function. $\newline$ The inner function is $-9x^{5}$ . The derivative of $-9x^{5}$ with respect to $x$ is $-45x^{4}$ (using the power rule: $\frac{d}{dx} [x^n] = nx^{n-1}$ ).
Apply chain rule: Apply the chain rule. $\newline$ Now, we multiply the derivative of the outer function by the derivative of the inner function. This gives us $(\frac{1}{(-9x^{5})}) \times (-45x^{4})$ .
Simplify expression: Simplify the expression. $\newline$ Simplifying the expression $\frac{1}{(-9x^{5})} \cdot (-45x^{4})$ gives us $-\frac{45x^{4}}{-9x^{5}}$ which simplifies to $\frac{5}{x}$ .
Include base conversion derivative: Include the derivative of the base conversion. $\newline$ We must not forget to include the denominator $\ln(8)$ from the base conversion in Step $2$ . The final derivative of the function $y$ with respect to $x$ is $\frac{5}{x}/\ln(8)$ .
Write final answer: Write the final answer. $\newline$ The final answer for the derivative of the function $y=\log_{8}(-9x^{5})$ is $y' = \frac{5}{x}\cdot\frac{1}{\ln(8)}$ .