Write down the equation that needs to be differentiated. For example, differentiate f(x) = e^(2x).
Identify the general rule for differentiating the natural exponential e, which is given as (d/dx) e^x = e^x. The derivative of e^x is itself.
Apply the rule to the nested function of the general type e^(ax) where (a) is a real number. In these problems, there are essentially two functions: the outer function as e^ax and the nested function of (ax). The rule is that the derivative of f(x) = e^(ax) for some real number (a) is f'(x) = (d/dx)(ax)*(d/dx)e(ax); thus, the derivative of e^(ax) is itself, multiplied by the derivative of the exponential value (ax), which is (a).
Apply the rules to the equation. Using the example, the derivative of e^2x is the derivative of the exponential variable (2x) multiplied by the derivative of expression itself (e^2x). It is seen as:
F(x) = e^(2x)
F'(x) = 2e^(2x)
Write down the equation that needs to be differentiated. For example, differentiate f(x) = ln (3x).
Identify the general rule for differentiation of a natural log, which is given as (d/dx) ln(x) = 1/x. The derivative of ln(x) is 1/x.
Apply the rule to the nested function of ln (ax) where (a) is a real number. As with the exponential function, if there is a nested equation (ax) within the equation ln (ax), the derivative of both the nested and whole equation must be evaluated. Thus, the derivative of the general form ln (ax) is the derivative of the whole function [(d/dx) ln(ax) =1/ax] multiplied by the derivative of the nested function [(d/dx) ax = a], giving the result as f'(x) = a / ax.
Apply both rules to the function to be differentiated. Using f(x) = ln (3x), differentiation of the outer function (ln (3x)) multiplied by the inner or nested function (3x) gives the result of f'(x) = 3/(3x). In this particular case, the 3 values cancel, resulting in a final answer of f'(x) = 1/x.