Derivative Rules

d/dx sin(x)

cos (x)

d/dx cos (x)

-sin(x)

d/dx tan (x)

sec^² (x)

d/dx sec (x)

sec(x)tan(x)

d/dx csc (x)

-cscx(x)cot(x)

d/dx cot(x)

-csc² (x)

d/dx e^x

e^x u’

d/dx b^x

b^x ln(b) u’ or e^ln(b)x ln(b)

Conditions of Critical Numbers

f’( c ) = 0 f’( c )= DNE and c must be in domain

Inflection point from equation

f’’ (x)= 0

conditions for differential equations

• If f is differentiable, then f is continuous

• If f is continuous, then f is not always differentiable

• Jump discontinuity

• Sharp corners

• Vertical tangent

• Vertical asymptote

Linearization/Tangent Line Approximation

f(a)+f’(a)(x-a)