Find the derivative of the following function. y=log_(3)(3x^(2)+8x) Answer: y^(')=

Recognize Function Type: Recognize that the function y = log_3(3x^2 + 8x) is a logarithmic function with a base other than e. To find the derivative, we will use the change of base formula and the chain rule.Apply Change of Base: Apply the change of base formula to rewrite the function in terms of the natural logarithm (ln). y = log_3(3x^2 + 8x) can be rewritten as y = ln(3x^2 + 8x) / ln(3).Use Chain Rule: Differentiate the function using the chain rule. The derivative of ln(u) with respect to x is 1/u * du/dx, where u is a function of x. Let u = 3x^2 + 8x, then du/dx = 6x + 8.Compute Derivative: Compute the derivative of y with respect to x. y' = d/dx [ln(3x^2 + 8x) / ln(3)] = 1/ln(3) * d/dx [ln(3x^2 + 8x)] y' = 1/ln(3) * (1/(3x^2 + 8x)) * (6x + 8)Simplify Result: Simplify the derivative. y' = (6x + 8) / ((3x^2 + 8x) * ln(3))

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Find derivatives of logarithmic functions

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Q. Find the derivative of the following function.

\newline

y=\log _{3}\left(3 x^{2}+8 x\right)

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Answer:

y^{\prime}=

Recognize Function Type: Recognize that the function $y = \log_3(3x^2 + 8x)$ is a logarithmic function with a base other than $e$ . To find the derivative, we will use the change of base formula and the chain rule.
Apply Change of Base: Apply the change of base formula to rewrite the function in terms of the natural logarithm (ln). $\newline$ $y = \log_3(3x^2 + 8x)$ can be rewritten as $y = \frac{\ln(3x^2 + 8x)}{\ln(3)}$ .
Use Chain Rule: Differentiate the function using the chain rule. The derivative of $\ln(u)$ with respect to $x$ is $\frac{1}{u} \cdot \frac{du}{dx}$ , where $u$ is a function of $x$ . $\newline$ Let $u = 3x^2 + 8x$ , then $\frac{du}{dx} = 6x + 8$ .
Compute Derivative: Compute the derivative of $y$ with respect to $x$ . $y' = \frac{d}{dx} \left[\frac{\ln(3x^2 + 8x)}{\ln(3)}\right] = \frac{1}{\ln(3)} \cdot \frac{d}{dx} \left[\ln(3x^2 + 8x)\right]$ $y' = \frac{1}{\ln(3)} \cdot \left(\frac{1}{3x^2 + 8x}\right) \cdot (6x + 8)$
Simplify Result: Simplify the derivative. $y' = \frac{6x + 8}{(3x^2 + 8x) \cdot \ln(3)}$