Calculus derivative test

derivative of e^x

e^x

derivative of a^x

a^x *ln(a)

derivative of ln x

1/x

derivative of log _a x

1/(x* ln a)

derivative of arcsin x

1/(root(1-x²))

derivative of arccos x

-1/(root(1-x²))

derivative of arctan x

1/(1+x²)

derivative of arccot x

-1/(1+x²)

derivative of arcsec x

1/(|x|* root(x²-1))

derivative of arccsc x

-1/(|x|* root(x²-1))

How to incorporate chain rule into the arc derivatives

replace the x with u, and numerator with the derivative of u

How to incorporate chain rule into the a^x derivative

change the x to u, and multiply the a^x derivative by the derivative of u

How to incorporate chain rule into the log_a x derivative

change x to u and multiply log_a x derivative by the derivative of u

Steps to find Inverse of a function

1. use first derivative test to show that function is monotonic

2. switch x and y in the function

3. solve for y

4. state that the domain for f(x) is the range for f^-1(x)

5. verify that f^-1(f(x))=f(f^-1(x))=x

If F is increasing on its domain then

f^-1 is

increasing on its domain

If f and g are inverses than when f(input)=output, g’(output)=

(1/f’(input)

If f(a)=b, then f^-1(b)=

a

In inverse trig functions, y represents an ___, while x represents a ____

angle, triangle ratio

Find derivative of an arc trig function implicitly means:

use the derivative of the non inverse function = to dy/dx under 1, then use the right triangle to find the measurements

How to find derivative of: y= arcsin x

sin y =x, so the derivative of that is (cos y)* (dy/dx)=1

Turn that to (1/cos y)=dy/dx to find dy/dx alone and plug in numbers from the right triangle to get answer.