The document provides an overview of derivatives, defining them as the instantaneous slope of a function and outlining various notations for representation. It details methods for finding derivatives, including the power rule, product rule, quotient rule, and chain rule, along with examples for each. Additionally, it highlights important trigonometric derivatives and general rules for differentiation.

1. Derivatives
What they are and how do we find them?

2. Definition of a Derivative
The instantaneous slope of a
function/the slope of a
function at a specific instant

3. Notations of Derivative of y=f(x)
f’(x)
y’
dy/dx
d/dxf(x)

4. Numerical Derivative
f’(a) = lim f(x)-f(a)
x->a x-a
f’(x) = lim f(x+h)-f(x)
h->0 h

5. Finding a Derivative
Tips and Tricks

6. Trigonometric Rules
d/dxsin(x) = cos(x)
d/dxcos(x) = -sin(x)
d/dxtan(x) = sec2(x)
d/dxsec(x) = sec(x)tan(x)
d/dxcot(x) = -csc2(x)
d/dxcsc(x) = -csc(x)cot(x)

7. Other Important Derivatives to Know
d/dx ex = ex
d/dx ax = (ax) ln(a)
d/dx ln(x) = 1/x

8. Power Rule
If f(x)=xn,
then f’(x)=nxn-1.

9. Power Rule Examples
f(x)=x3  f’(x)=3x2
f(x)=x5  f’(x)=5x4

10. Product Rule
If f(x)=gh,
then f’(x)=g’h+h’g.

11. Product Rule Examples
f(x) = x2sin(x)  f’(x) = 2xsin(x)+x2cos(x)
f(x) = 3x3ex  f’(x) = 9x2ex+3x3ex

12. Quotient Rule
If f(x)=u/v, then
f’(x)=vu’-u’v
v2.

13. Quotient Rule Examples
f(x) = x2/sin(x)  f’(x) = sin(x)2x-x2cos(x)
x4
f(x) = sin(x)/ln(x)  f’(x) = ln(x)cos(x)-sin(x)*(1/x)
(ln(x))2

14. Chain Rule
If f(x)=g(h(x)), then
f’(x)=g’(h(x))*h’(x)

15. Chain Rule Examples
f(x) = sin(ln(x))  f’(x) = cos(ln(x))*(1/x)
f(x) = e(x^2)  e(x^2)*2x