AP Calculus BC - Derivatives Formulas

Limit definition of derivative

lim x→0 f(x+h) - f(x) / h

lim x→c f(x) - f(c)/x -c

Average rate of change

f(b) - f(a)/b-a

Slope of the secant line

Instantaneous rate of change

Derivative (f’(c))

Slope of tangent line

Derivatives fail to exist

1. If f is not continuous at x=c, then f'(c) is undefined.

2. Cusp (lim x->c f'(x) ≠ x->c f'(x))

3. Vertical tangent line, dy/dx = undefined

Normal line

Perpendicular to tangent line

Critical points

f’(x) = 0/undefined

Constant rule

d/dx [c] = 0

Constant multiple rule

d/dx [cf(x)] = cf’(x)

Power rule

d/dx [xⁿ] = nxⁿ−¹

d/dx [x^u] = u’x^(u-1)

Sum rule/difference rule

d/dx [f(x)±g(x)] = f’(x) ± g’(x)

Product rule

d/dx [f(x)*g(x)] = fg’+gf’

Quotient rule

d/dx [f(x)/g(x)] = gf’-fg’/[g²]

Chain rule

d/dx [f(g(x))] = f’(g(x))g’(x)

d/dx [e^x]

e^x

u’e^u

d/dx [a^x]

a^xlna

a^ulna*u’

d/dx [lnx]

1/x

u’/u

d/dx [log_ax]

1/xlna

u’/ulna

d/dx [sinx]

cosx

u’cosu

d/dx [cosx]

-sinx

u’*-sinu

d/dx [tanx]

sec²x

u’(sec²u)

d/dx [secx]

secxtanx

u’(secutanu)

d/dx [cscx]

-cscxcotx

u’(-cscucotu)

d/dx [cotx]

-csc²x

u’(-csc²u)

d/dx [arcsinx]

1/√1-x²

u’/√1-u²

d/dx [arccosx]

-1/√1-x²

-u’/√1-u²

d/dx [arctanx]

1/1+x²

u’/1+u²

d/dx [arccotx]

-1/1+x²

-u’/1+u²

d/dx [arcsecx]

1/|x|√x²-1

u’/|u|√u²-1

d/dx [arccscx]

-1/|x|√x²-1

-u’/|u|√u²-1

L’hospital rule

lim x→a f(x)/g(x)=0/0 or ∞/∞, then lim x→a f’(x)/g’(x)

Mean Value Theorem

1. f(x) is continuous on [a,b] and differentiable on (a,b)

2. f(b)-f(a)/b-a = N

3. Therefore, by MVT there must exist be some value “c” such that f’(c) = N on (a,b)

Rolle’s Theorem

1. f(x) is continuous on [a,b] and differentiable on (a,b)

2. f(a) = f(b)

3. Therefore, by Rolle’s there must be some value “c” such that f’(c) = 0 on (a,b)