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g(x)=-log_(5)(-x-2)+3

$g(x)=-\log _{5}(-x-2)+3$

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Find derivatives of logarithmic functions

Full solution

Q.

g(x)=-\log _{5}(-x-2)+3

Identify Function: Identify the function to differentiate. $\newline$ We need to find the derivative of $g(x) = -\log_{5}(-x-2) + 3$ with respect to $x$ .
Apply Chain Rule: Apply the chain rule to the logarithmic function. $\newline$ The derivative of $\log_b(u)$ with respect to $x$ is $\left(\frac{1}{u \ln(b)}\right) \cdot \frac{du}{dx}$ , where $u$ is a function of $x$ . Here, $b = 5$ and $u = -x - 2$ .
Differentiate Inner Function: Differentiate the inner function $u = -x - 2$ . $\newline$ The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = -1$ .
Combine Results: Combine the results from Step $2$ and Step $3$ . $\newline$ The derivative of $-\log_{5}(-x-2)$ is $-1 \times \left(\frac{1}{((-x - 2) \cdot \ln(5))}\right) \times (-1) = \frac{1}{((-x - 2) \cdot \ln(5))}.$
Differentiate Constant Term: Differentiate the constant term. $\newline$ The derivative of a constant is $0$ , so the derivative of $+3$ is $0$ .
Combine Derivatives: Combine the derivatives of all parts of the function. $\newline$ $g'(x) = \frac{1}{(-x - 2) \cdot \ln(5)} + 0$
Simplify Derivative: Simplify the derivative if possible. $\newline$ The derivative is already in its simplest form, so $g'(x) = \frac{1}{(-x - 2) \cdot \ln(5)}$ .

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