Derivative

Basic Derivative

100

What is the definition of a derivative. 

the slope of a tan line or average RoC (slope) 

200

What is the notation for a derivative.

F'(x) or f prime of x

300

20x^2

40x

400

5x^3 + 10x^2 + 50x

15x^2 + 20x + 50

500

57x^7-23x^5-15x^3+8x+2

399x^6-115x^4-45x^2+8

Chain Rule

100

f(x)=(2x^2 +5)^5

20xx(2x^2 +5)^4

200

f(x)=(ln(x^2-1))

f'(x)=2x/x^2-1

300

f(x)=4(1+7x)^9

252(1+7x)^8

400

b(y)=(y^3-5)^-4

B'(y)=-12y^2/(y^3-5)^5

500

f(x)=(2x-3)^4 x (x^2+x+1)^5

f'(x)=8(2x-3)^3 x (x^2+x+1)+(2x^3-x^2-x-3)64 x (10x+5)

Product Rule

100

f(x)=(x^2-1)x(x+5)

f'(x)=3x^2+10x-1

200

y=2(6-x^2)^5

y'=10(6-x^2)^4x(-2x)

300

x^2ln(x)

x(2lnx+1)

400

f(x)=(x^2-2)(x^-1+2)

f'(x)4x+2x^-2+1

500

(2x^4-2x^2+2)(5x^4+1)

8x^7-60x^5+48x^3-4x

Quotient Rule

100

y=x^2/x^2-3

dy/dx=-6x/(x^2-3)^2

200

y=2-x/3x+1

dy/dx=-7/(3x+1)^2

300

y=2/(5x+1)

y'=-30/(5x+1)-4

400

2x^3/4-x

24x^2-4x^3/(4-x)^2

500

y=x^2 +x-2/x^3 + 6

-x^4 - 2x^3 + 6x^2 + 12x+6/(x^3 +6)^2

More examples (mixed)

100

4x^5

20x^4

200

h(t)=2/2t^2-3 

h'(t)=-8t/4t^4-12t^2+9

300

f(x)=(1+x^-4)(x^5 +5)

5x^4+1- 20/x^5

400

sqrt(5x^3)/5x^3

-3/2x^5/2sqrt(5)

500

h(u)=sqrt(u-1) x sqrt3(2u+3)

10u+5/6sqrt(u-1) x (2u+3)^2/3