Calculus Derivatives

Chain Rule

100

y = (x+1)³

y' = 3(x+1)²

200

f(x) = (3t+6)^2/3

f'(x) = 2 / (3t+6)^1/3

300

f(x) = -2(9x²+8)^1/2

f'(x) = -18x / (9s²+8)^1/2

400

f(x) = -2(9s²+8)^1/2

f'(x) = -18s / (9s²+8)^1/2

500

y = x²(49-x²)^1/2

y' = (98x-3x³) / (49-x²)^1/2

Product and Quotient Rule

100

y = (x²+3) (x³-9)

y' = 5x⁴+9x²-18x

200

f(x) = x / (x-6)

f'(x) = -6 / (x-6)²

300

y = (5x-2) / (x²+1)

y' = (-5x²+4x+5) / (x²+1)²

400

f(x) = (2x+5) (x-8)^1/2

f'(x) = (6x-27) / 2(x-8)^1/2

500

f(x) = x⁷ ( 1-(1 / (x+4) )

f'(x) = (7x⁸+50x⁷+84x⁶) / (x+4)²

Logarithmic

100

y = ln(x)^1/4

y' = 1 / 4x

200

f(x) = 2x²ln(2x)

f'(x) = 4xln(2x)+2x

300

y = ln(x+ (8+x²) )

y' = 1 / (8+x²)^1/2

400

f(x) = log₂(1-4x)

f'(x) = -4 / (1-4x)ln(2)

500

y = ln( (3t+1)⁴ / (2t-1)⁵ )

y' = (-6t-22) / (3t+1) (2t-1)

Exponential

100

y = 9e^x+6

y' = 9e^x

200

f(x) = e¹⁸

y' = 0

300

y = x4^x

y' = 4^x+x4^xln(4)

400

f(x) = e^-4/t5

f'(x) = (20 / t⁶) (e^-4/t5)

500

y = 9 / (e^x+e^-x)

y' = (-9(e^x-e^-x) ) / (e^x+e^-x)²

Trigonometric

100

f(x) = -cos(x)

f'(x) = sin(x)

200

y = x²tan(x)

y' = x²sec²(x) + 2xtan(x)

300

y = 4csc(2x-1)

y' = -8cot(2x-1)csc(2x-1)

400

f(x) = (tan(x)-1) / sec(x)

f'(x) = (1+tan(x)) / sec(x)

500

y = 2x / (1-cot(x))

y' = (2-2cot(x)-2xcsc²(x)) / (1-cot(x))²