Differentiation

Power Rule/Exponentials

100

f(t) = t³ - 3t² + 8t - 4

f'(t) = 3t² - 6t + 8

200

y = 4x^3/2 - 5x^1/2

y' = 6x^1/2 - 5/2x^-1/2

300

z = ln(4)4^x

z' = (ln(4))²4^x

400

f(t) = e^(t+2)

f'(t) = e^(t+2)

500

f(x) = e^pi+pi^x

f'(x) = ln(pi)pi^x

Product Rule

100

f(x) = xe^x

f'(x) = e^x(1+x)

200

y = (t²+3)e^t

y' = (t²+2t+3)e^t

300

y = x³ln(x)

y' = x²(3ln(x)+1)

400

f(t) = (t³-7t²+1)e^t

f'(t) = (t³-4t²-14t+1)e^t

500

y = 5x² + sin(x)cos(x)

y' = 10x + cos²(x) - sin²(x)

Quotient Rule

100

f(x) = x/e^x

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

200

z(t) = (3t+1)/(5t+2)

z'(t) = 1/(5t+2)²

300

y = (x²)/(3x-1)

y' = (x(3x-2)) / (3x-1)²

400

y = (4sin(x)) / (2x+cos(x))

y' = (8xcos(x)-8sin(x)+4) / (2x+cos(x))²

500

g(x) = (1+ln(x)) / (x²-ln(x))

g'(x) = (1-x²-2x²ln(x)) / (x(x²-ln(x))²)

Chain Rule

100

w = 100e^-x2

w' = -200xe^-x2

200

y = (1-4x+7x⁵)³⁰

y' = 30(35x⁴-4)(1-4x+7x⁵)²⁹

300

y = 3tan(x^1/2)

y' = (3sec²(x^1/2)) / (2x^1/2)

400

f(y) = e^{e^(y^2)}

f'(x) = 2ye^{(e^(y^2)+y^2)}

500

y = cos²(x³)

y' = -6x²cos(x³)sin(x³)

Implicit Differentiation

100

xy + x + y = 5

dy/dx = (-y-1) / (x+1) 

200

x³ + y³ = 4

dy/dx = -x²/y²

300

ln(x) + ln(y²) = 3

dy/dx = -y/2x

400

sin(xy) = 2x + 5

dy/dx = (2 - ycos(xy)) / (xcos(xy))

500

e^xy = e^4x - e^5y

(4e^4x - ye^xy) / (xe^xy + 5e^5y)