Calculus Chapter 3

Product Rule

100

find the derivative of (x²+1)(x³+3)

5x⁴+3x²+6x

200

Find the derivative of (4t-t)(t³-8t+12)

20t⁴-132t³+24t²+96t-12

300

(1+x^1.5)(x^-3-2x^0.3333)

-3x^-4-1.5x^-2.5-0.666x^-0.666-3.666x^0.8333

Quotient Rule

100

(x²-1) / (x²+1)

4x / (x²+1)²

200

6x² / (2-x)

24x-6x² / (2-x)²

300

(3w+w⁴) / (2w²+1)

4w⁵ + 4w³ -6w² + 3  / (2w² + 1)²

Multiple derivatives

100

The position of a particle moving along a straight line is given by s=t³-6t²+12t-8

Find the velocity

3t²-12t+12

200

Aparticle moves along a line such that its position s(t)= 2t³-9t²+12t-4 for t>or=0

Find the acceleration of the function  

s"(t)= 12t-18

300

A particle moves along a line such that its position is S(t)= t⁴-4t³

Find the particle's jerk

24t-24

Chain Rule

100

differentiate the function with respect to x

y= (5x²+3)⁴ 

40x(5x²+3)³ 

200

Differentiate the function with respect to x

f(x)= (-2x⁴+5)^0.3333

8x³ / 3(-2x⁴+5)^0.6666

300

Differentiate the function 

x(t)=cos(t²+1)

-2tsin(t²+1)

Implicit Differentiation

100

Find y′ by implicit differentiation

2y³ + 4x² -y = x⁶

6x⁵-8x / 6y²-1

200

Find y′ by implicit differentiation 

7y² + sin(3x) = 12-y⁴

y'=-3cos(3x) / 14y + 4y³

300

Find y′ by implicit differentiation

tan(x²y⁴)=3x+y²

3-2xy⁴sec²(x²y⁴) / 4x²y³sec²(x²y⁴)-2y