Math Problem Statement

Find the derivative of the function. f(x) = log5(x) f '(x) =

Solution

To find the derivative of f(x)=log5(x)f(x) = \log_5(x), we can use the change of base formula for logarithms along with the derivative rules.

  1. Rewrite log5(x)\log_5(x) in terms of the natural logarithm: f(x)=log5(x)=ln(x)ln(5)f(x) = \log_5(x) = \frac{\ln(x)}{\ln(5)}

  2. Differentiate f(x)f(x) with respect to xx: f(x)=1ln(5)ddx(ln(x))f'(x) = \frac{1}{\ln(5)} \cdot \frac{d}{dx}(\ln(x))

  3. Use the derivative of ln(x)\ln(x), which is 1x\frac{1}{x}: f(x)=1ln(5)1x=1xln(5)f'(x) = \frac{1}{\ln(5)} \cdot \frac{1}{x} = \frac{1}{x \ln(5)}

So, the derivative of f(x)=log5(x)f(x) = \log_5(x) is: f(x)=1xln(5)f'(x) = \frac{1}{x \ln(5)}

Would you like further clarification on any step?


Related Questions

  1. What is the derivative of f(x)=loga(x)f(x) = \log_a(x) for an arbitrary base aa?
  2. How do we differentiate f(x)=ln(kx)f(x) = \ln(kx), where kk is a constant?
  3. What is the derivative of g(x)=elog5(x)g(x) = e^{\log_5(x)}?
  4. How would you find the second derivative of f(x)=log5(x)f(x) = \log_5(x)?
  5. How is the change of base formula for logarithms useful in calculus?

Tip

When differentiating logarithmic functions with arbitrary bases, converting them to natural logarithms simplifies the process!

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Math Problem Analysis

Mathematical Concepts

Calculus
Logarithmic Differentiation

Formulas

Change of base formula: log_a(x) = ln(x)/ln(a)
Derivative of natural logarithm: d/dx(ln(x)) = 1/x

Theorems

Change of Base Formula
Derivative of Logarithmic Functions

Suitable Grade Level

Grades 11-12