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

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

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Full solution

Q. g(x)=−log⁡5(−x−2)+3 g(x)=-\log _{5}(-x-2)+3
  1. Identify Function: Identify the function to differentiate.\newlineWe need to find the derivative of g(x)=−log⁡5(−x−2)+3g(x) = -\log_{5}(-x-2) + 3 with respect to xx.
  2. Apply Chain Rule: Apply the chain rule to the logarithmic function.\newlineThe derivative of log⁡b(u)\log_b(u) with respect to xx is (1uln⁡(b))⋅dudx\left(\frac{1}{u \ln(b)}\right) \cdot \frac{du}{dx}, where uu is a function of xx. Here, b=5b = 5 and u=−x−2u = -x - 2.
  3. Differentiate Inner Function: Differentiate the inner function u=−x−2u = -x - 2.\newlineThe derivative of uu with respect to xx is dudx=−1\frac{du}{dx} = -1.
  4. Combine Results: Combine the results from Step 22 and Step 33.\newlineThe derivative of −log⁡5(−x−2)-\log_{5}(-x-2) is −1×(1((−x−2)⋅ln⁡(5)))×(−1)=1((−x−2)⋅ln⁡(5)).-1 \times \left(\frac{1}{((-x - 2) \cdot \ln(5))}\right) \times (-1) = \frac{1}{((-x - 2) \cdot \ln(5))}.
  5. Differentiate Constant Term: Differentiate the constant term.\newlineThe derivative of a constant is 00, so the derivative of +3+3 is 00.
  6. Combine Derivatives: Combine the derivatives of all parts of the function.\newlineg′(x)=1(−x−2)⋅ln⁡(5)+0g'(x) = \frac{1}{(-x - 2) \cdot \ln(5)} + 0
  7. Simplify Derivative: Simplify the derivative if possible.\newlineThe derivative is already in its simplest form, so g′(x)=1(−x−2)⋅ln⁡(5)g'(x) = \frac{1}{(-x - 2) \cdot \ln(5)}.

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