Derivatives of Exponential and Logarithmic Functions

The natural exponential function f(x) = ex is unique because its derivative is itself.

Formula: d/dx (ex) = ex

For a general exponential function f(x) = ax, the derivative involves the natural logarithm of the base 'a'.

Formula: d/dx (ax) = ax * ln(a)

Note that the natural exponential case (a=e) fits this formula, as ln(e) = 1, giving d/dx (e^x) = e^x * 1 = e^x.

The derivative of the natural logarithm function f(x) = ln(x) is one over x. Note that ln(x) is only defined for x > 0.

Formula: d/dx (ln x) = 1 ÷ x (for x > 0)

For a general logarithm function with base 'a' (f(x) = loga(x), where a > 0, a ≠ 1, x > 0), you can use the change of base formula (loga(x) = ln(x) ÷ ln(a)) and the constant multiple rule.

Formula: d/dx (loga x) = 1 ÷ (x * ln a) (for x > 0)

You must use the Chain Rule. The general form is: differentiate the outer function, evaluate it at the inner function, and multiply by the derivative of the inner function.

• **For e^g(x):** d/dx (eg(x)) = eg(x) * g'(x)
• **For a^g(x):** d/dx (ag(x)) = ag(x) * ln(a) * g'(x)
• **For ln(g(x)):** d/dx (ln(g(x))) = 1 ÷ g(x) * g'(x) = g'(x) ÷ g(x) (for g(x) > 0)
• **For log_a(g(x)):** d/dx (loga(g(x))) = 1 ÷ (g(x) * ln a) * g'(x) = g'(x) ÷ (g(x) * ln a) (for g(x) > 0)

Remember to always apply the Chain Rule when the argument of the exponential or logarithmic function is something other than a simple variable like 'x'.