Derivatives

Sum/Difference Rule

100

f(x) = 4x³+2x²−3x+1

f'(x) = 12x²+4x-3

200

y = 4x²+1

8x

300

What are the formulas for Sum Rule and Difference Rule?

Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)

Difference Rule: d/dx [f(x) - g(x)] = f'(x) - g'(x)

400

y = -4x²-5x-2

y' = -8x-5

500

y = 17x⁵+12x⁴-7x³-15x²-x+235

y' = 85x⁴+48x³-21x²-30x

Power Rule

100

y= x⁷

y’= 7x⁶

200

y= 3/x⁷

y’= -21/x⁸

300

What's the formula for Power Rule?

d/dx (xⁿ) = nx^n-1

400

y= x²( 2x² - 3x)

y’= 8x³ - 9x²

500

f(x)= x^1/2 - 6x^1/3

f’(x)= 1/2x^1/2 - 2/x^2/3

Product Rule

100

y = 4x³(9x⁵)

y' = 288x⁷

200

y = 2x³(-x³+3)

y' = 18x⁵-18x²

300

What's the formula for Product Rule?

Product rule: d/dx [f(x) * g(x)] = f'(x)*g(x) + f(x)*g'(x)

400

  y = (-4x⁴ +1) (-4x⁴)  

y' = 128x⁷ -16x³

500

  y = (-5x⁴ -3x³ -2) (3x³ -2)

y' = -105x⁶ - 54x⁵ +40x³

Quotient Rule

100

y=2/x⁵-5

-10x⁴/(x⁵-5)²

200

y = x³/(5x²-4)

y' = (5x⁴-12x²)/(5x²-4)²

300

What's the formula for Quotient Rule?

Quotient Rule: d/dx [f(x)/g(x)] = (f'(x)*g(x) - f(x)*g'(x))/g(x)^2

400

y = (x⁴+4x²-4)/(2x³-4)

y' = (2x⁶-24x⁴+24x²-32x)/(2x³-4)²

500

y= 4x⁴-4x²+5/2x^5/3+3

y'=(56/3x^14/3 - 8/3x^8/3+48x³ - 24x - 50/3x^2/3)/(2x^5/3+3)²

Chain Rule

100

y = (x³+3)⁵

y' = 15x²(x³+3)⁴

200

y = (-5x³-3)³

y' = -45x²(-5x³-3)²

300

What's the formula for Chain Rule?

Chain Rule: d/dx [f(g(x)) = f'(g(x) * g'(x)

400

f(x) = (-2x²+1)^1/2

f'(x) = (-2x)(-2x²+1)^-1/3

500

f(x) = (-2x⁴+5)^1/3

f'(x) = (-8/3 x³)(-2x⁴+4)^-2/3