AP Calculus Unit 3 Formulas

Derivative of lnx

d/dx[lnx]=1/x, d/dx[IxI]=1/x

Derivative of General Logarithmic Functions

d/dx[logbx]=1/xlnb, d/dx [logbIxI]=1/xlnb

Derivative of Inverse Secant

d/dx[sec^-1x]=1/IxIsqrtx^2-1

Derivative of Inverse Sine

d/dx[sin^-1x]=1/sqrt1-x^2

Derivative of Inverse Tangent

d/dx[tan^-1x]=1/1+x^2

Chain Rule

dy/dx=(dy/du)(du/dx) or d/dx[f(g(x))]=f'(g(x))g'(x)

Derivative of the Inverse Functions

(f^-1)'(yo)=1/f'(xo) where yo=f(xo) *(xo,yo) is a point on the original function, (yo,xo) is a point on the inverse function

Logarithmic Differentiation

*see notes for examples

Guidelines for Implicit Differentiation

*use when equation cannot be written a y=?, 1. Differentiate both sides of the equation with respect to x 2. Collect all terms involving y' on the left side and all terms not with y' on the right 3. Factor out y' of thenleft side equation 4. Solve for y'

Derivatives of General Exponential Functions

If b>0, then for all x: d/dx [b^x]=b^xlnb, d/dx[e^x]=e^x

Derivative of Inverse Cosine

d/dx[cos^-1x]=-1/sqrt1-x^2

Derivative of Inverse Cotangent

d/dx[cot^-1x]=-1/1+x^2

Derivative of Inverse Cosecant

d/dx[csc^-1x]=-1/IxIsqrtx^2-1