The equation of the tangent line is
In this section we compute derivatives involving and .
1 Derivative of
We begin by computing the derivative of the exponential function . Recall that is a number whose value is approximately 2.72. Like and , is an irrational number, meaning that its decimal representation is non-terminating and non-repeating. The significance of the number in mathematics is underscored by the simplicity of the differentiation formula we are about to discover. The result hinges on a limit that we analyzed in the Numerical Limits section:
The derivative of is found as follows:
Thus the derivative of is itself!
The point of tangency is since . The derivative is , so the slope of the tangent line is . Using the point-slope formula, we can make the equation of the tangent line: which can be rewritten in slope-intercept form as
The equation of the tangent line is
2 Derivative of
Next, we use the definition of the derivative to find the derivative of the natural logarithm.
We will need the special limit
We can use a table of values to verify this limit: and Now we are prepared to use the definition to find the derivative of :
If then
Note that we made the substitution so that and also note that is equivalent to . To recap, which gives a memorable mathematical relationship between the transcendental, natural logarithm function the algebraic, reciprocal function.
In the formula , the domains of the functions do not match. The domain of is all non-zero , but the domain of is only . This leads us to ask if there is a function defined for whose derivative is . The answer is yes and the function is . In other words, This result follows from the definition of the derivative exactly as it was used above. The absolute value bars will eventually drop in the computation because becomes positive as .
The slope of the tangent line is and since the derivative is , the slope is .