Calculus 1

Power Rule

d/dx[x^n] = nx^(n-1)

Constant Rule

d/dx[c] = 0

Constant Multiple Rule

d/dx[c·f(x)] = c·f'(x)

Sum/Difference Rule

(f ± g)' = f' ± g'

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

sec²(x)

Derivative of csc(x)

-csc(x)cot(x)

Derivative of sec(x)

sec(x)tan(x)

Derivative of cot(x)

-csc²(x)

Derivative of e^x

e^x

Derivative of ln(x)

1/x

Product Rule

(fg)' = f'g + fg'

Quotient Rule

(f/g)' = (gf' - fg')/g²

Average Rate of Change

[f(b) - f(a)]/(b - a)

Instantaneous Rate of Change

f'(a); slope of the tangent line at x = a

Tangent Line Equation

y - y₁ = m(x - x₁), where m = f'(a)

Derivative Definition (h form)

f'(a) = lim(h→0) [f(a+h) - f(a)]/h

Derivative Definition (x form)

f'(a) = lim(x→a) [f(x) - f(a)]/(x-a)

Differentiability Implies Continuity?

True

Continuity Implies Differentiability?

False

Three Reasons a Derivative May Not Exist

Discontinuity, sharp corner, vertical tangent

Continuity at x = a

lim(x→a) f(x) = f(a)

Differentiability at x = a

Left and right derivatives exist and are equal

Derivative Notations

f'(x), y', and dy/dx

Derivative of a Negative Power

d/dx[x^(-n)] = -n·x^(-n-1)

Derivative Estimate From a Table

Use [f(b) - f(a)]/(b-a) with nearby points around a

Derivative of a Constant Times a Function

Keep the constant and differentiate the function