Derivative of Trig Functions, Exponentials, and Inverses

d/dx[tan^-1]

1/(1+x²)

d/dx[cot^-1]

-1/(1+x²)

d/dx[sec^-1]

1/x(√x²-1)

d/dx[csc^-1]

-1/x(√x²-1)

d/dx[sin^-1]

1/(√1-x²)

d/dx[cos^-1]

-1/√1-x²

d/dx[sin(x)]

cos(x)

d/dx[cos(x)]

-sin(x)

d/dx[tan(x)]

sec²(x)

d/dx[cot(x)]

-csc²(x)

d/dx[sec(x)]

sec(x) * tan(x)

d/dx[csc(x)]

-csc(x) * cot(x)

d/dx[e^x]

e^x

d/dx[b^x]

ln(b) * b^x

d/dx[e^f(x)]

f’(x) * e^f(x)

d/dx[b^f(x)]

ln(b) * f’(x) * b^f(x)

a. lim_x→0(e^x-1)/x =

a. 1

lim_x→0(b^x-1)/x =

ln(b)

d/dx[ln(x)] for x>0

1/x

d/dx[ln(|x|)] for x/=0

1/x

d/dx[log_b(x)] for x>0

1/(ln(b)) * x

d/dx[log_b(|x|)] for x/=0

1/(ln(b)) * x

d/dx[ln(f(x))] for f(x)>0 with f’(x) defined

f’(x)/f(x)

d/dx[ln|f(x)|] for f(x)/=0 with f’(x) defined

f’(x)/f(x)

For Inverse Trig Function Derivatives:

Remember the original, make it negative for “co-”

For Trig Function Derivatives: If “co-” added,

Add a “co-” to original and make negative

b. lim_x→0sin(x)/x =

b. 1

lim_x→0cos(x)-1/x =

0

tan(x) =

sin(x)/cos(x)

sec(x) =

1/cos(x)

csc(x) =

1/sin(x)

cot(x) =

1/tan(x) = cos(x)/sin(x)

c. sin²(x) + cos²(x) = 

c. 1

sin(A+B) =

sin(A)cos(B) + cos(A)sin(B)

cos(A+B) =

cos(A)cos(B) - sin(A)cos(B)