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Derivatives of Logarithmic and Exponential Functions- DS2
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 if then
For an exponential function the graph of appears to be a scaled version of the original function. In particular, careful analysis of the graph of suggests that 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 To do so, we take advantage of the fact that we know the derivative of the natural exponential function, the inverse of In particular, we know that writing is equivalent to writing Now we differentiate both sides of this equation and observe that
The right hand side is simply by applying the chain rule to the left side, we find that
Next we solve for to get
Finally, we recall that so and thus
Natural Logarithm
For all positive real numbers