Related rates problems find how fast one quantity changes when you know how fast a related quantity changes. Write an equation that connects the quantities, differentiate both sides with respect to time t, then substitute the values at the moment you care about and solve for the unknown rate.
The derivatives of inverse trig functions are algebraic, with no trig left in them. The two you will use most are (arcsin x)′ = 1/√(1 − x²) and (arctan x)′ = 1/(1 + x²). Arccosine, arccotangent and arccosecant have the same derivatives as their partners, with a minus sign.
Logarithmic differentiation is a way to differentiate a function by taking the natural log of both sides first. Log properties turn products, quotients and powers into sums, which are easy to differentiate. It is the standard method for functions like x^x, where the variable is in both the base and the exponent.
The derivative of a logarithmic function starts with one rule: the derivative of ln x is 1/x, for x > 0. For another base, the derivative of log_a x is 1/(x ln a). With an inside function u, the chain rule gives (ln u)′ = u′/u: the derivative of the inside divided by the inside.
The derivative of an exponential function depends on its base. The derivative of e^x is e^x itself. For any other base a > 0, the derivative of a^x is a^x ln a. With an inside function u, use the chain rule: (e^u)′ = e^u · u′ and (a^u)′ = a^u ln a · u′.
The derivatives of trig functions are: the derivative of sin x is cos x, of cos x is −sin x, and of tan x is sec²x. The other three are (sec x)′ = sec x tan x, (csc x)′ = −csc x cot x and (cot x)′ = −csc²x. These formulas work only when x is in radians.
The derivative of an inverse function is the reciprocal of the original function’s derivative at the matching point. If f(a) = b, then (f⁻¹)′(b) = 1/f′(a), as long as f′(a) ≠ 0. You can find the slope of the inverse without finding a formula for the inverse.
Implicit differentiation finds dy/dx when y is not written by itself, as in x² + y² = 25. Differentiate both sides with respect to x, treating y as a function of x. Each time you differentiate a y term, multiply by dy/dx (the chain rule). Then solve for dy/dx.
The chain rule finds the derivative of a composite function, one function inside another. For y = f(g(x)), the derivative is f'(g(x)) · g'(x): take the derivative of the outside function, keep the inside the same, then multiply by the derivative of the inside.
The quotient rule finds the derivative of one function divided by another. For f(x)/g(x), the derivative is (f’g – fg’)/g²: the bottom times the derivative of the top, minus the top times the derivative of the bottom, all over the bottom squared. The order in the numerator matters.