MATH 220 Study

derivative of sin x

cos x

derivative of cos x

-sin x

derivative of tan x

sec²x

derivative of csc x

-csc x cot x

derivative of sec x

sec x tan x

derivative of cot x

-csc²x

what does it mean when an equation asks for the maximum height

the location where f’(x)=0

derivative of e^x

e^x

power rule

f(x) g(x)= f’(x) g(x) + g’(x) f(x)

power rule with more than 2 functions

f(x) g(x) h(x)= f’(x) g(x) h(x)+ f(x) g’(x) h(x) +f(x) g(x) h’(x)

quotient rule

f(x)/g(x)= f’(x)g(x)-g’(x)f(x)/G(x)²

chain rule

f(g(x))= f’(g(x)) g’(x)

derivative of an exponential function b^x

d/dx (b^x)=b^x ln(b)

e^ln(x)=?

x

chain rule

f’(x)=f(g(x))=f’(x) g(x)

applies to problems with base e example e^sinx

steps of finding the derivative of a logarithmic function

y=log b x

1) write as b^y=x

2) take derivative of both sides and use the b^x rule= b^x(ln(b))(x’)

• derivative written as b^y(ln(b))(y’)=1

• divide y’=1/(b^y)(ln(b))

3) sub x in for b^y from original equation

• y’=1/x ln(b) (answer!)

derivative of an exponential function

d/dx[b^x]=b^x (ln(b)) (x’)

taking derivatives of y=f(x)^(g(x))

steps

1) take natural log of both sides ln(y)=ln(f(x)^g(x))

2) use log rules and write it as

• ln(y)=g(x) ln(f(x))

3) take derivative of both sides

• 1/y (y’)= g’(x)(ln(f(x))+g(x)(ln(f(x))’

derivative of arcsinx

1/root (1-x²)

derivative of arccosx

-1/root(1-x²)

derivative of arctanx

1/x²+1