Derivatives of Logarithmic and Exponential Functions
Derivatives of Logarithmic and Exponential Functions
Objectives
- Find the derivative of logarithmic functions.
- Use logarithmic differentiation to determine the derivative of a function.
Summary
We have stated a rule for derivatives of exponential functions in the same spirit as the rule for power functions: for any positive real number \(a\text{,}\) if \(f(x) = a^x\text{,}\) then \(f'(x) = a^x \ln(a)\text{.}\)
For an exponential function \(f(x) = a^x\) \((a \gt 1)\text{,}\) the graph of \(f'(x)\) appears to be a scaled version of the original function. In particular, careful analysis of the graph of\(f(x) = 2^x\text{,}\) suggests that \(\frac{d}{dx}[2^x] = 2^x \ln(2)\text{,}\) which is a special case of the rule we stated in Section 2.1.
In what follows, we find a formula for the derivative of \(g(x) = \ln(x)\text{.}\) To do so, we take advantage of the fact that we know the derivative of the natural exponential function, the inverse of \(g\text{.}\) In particular, we know that writing \(g(x) = \ln(x)\) is equivalent to writing \(e^{g(x)} = x\text{.}\) Now we differentiate both sides of this equation and observe that
The righthand side is simply \(1\text{;}\) by applying the chain rule to the left side, we find that
Next we solve for \(g'(x)\text{,}\) to get
Finally, we recall that \(g(x) = \ln(x)\text{,}\) so \(e^{g(x)} = e^{\ln(x)} = x\text{,}\) and thus
Natural Logarithm
For all positive real numbers \(x\text{,}\) \(\frac{d}{dx}[\ln(x)] = \frac{1}{x}\text{.}\)