Derivatives of trig, exponential and log functions
The power rule handles x^3. What happens when the variable moves into an exponent, inside a sine, or inside a logarithm?
Where you are 3 inside
What this area is about
Growth, decay and anything that repeats are modeled by exponentials and trigonometric functions, and logarithms undo exponentials. None of these is a polynomial, so the power rule says nothing about them. This area finds their derivatives once, from the definition and the earlier rules, so they can be used like any other rule afterward.
How these fit together
All three are results, each a family of formulas with the argument behind it.
Exponentials. Derivatives of exponential functions shows that \displaystyle \frac{d}{dx}\big[a^x\big] = a^x \ln a for a > 0, so that e^x is its own derivative.
Trigonometric functions. Derivatives of the trigonometric functions finds \displaystyle \frac{d}{dx}\big[\sin x\big] = \cos x and \displaystyle \frac{d}{dx}\big[\cos x\big] = -\sin x, with x in radians, then gets the other four from the quotient rule.
Logarithms. Derivatives of logarithms and logarithmic differentiation differentiates e^y = x implicitly to get \displaystyle \frac{d}{dx}\big[\ln x\big] = \frac{1}{x} for x > 0, then uses the laws of logarithms to differentiate long products and powers such as x^x.
A path through it
No child formally requires another, but the lessons are written in the order exponentials, trigonometric functions, logarithms, and each refers back to the ones before. The trigonometric derivatives also need the limit \displaystyle \frac{\sin x}{x} \to 1 and the quotient rule; logarithms lean on e^x and implicit differentiation.
Next come L’Hôpital’s rule, antiderivatives of trigonometric functions and, much later, Taylor series.