G E 107C - Finding Derivatives

Chain Rule

100

f(x) = (3x+1)²

f'(x) = 6*(3x+1) = 18x + 6

200

f(x) = sin(4x²)

f"(x) = 8xcos(4x²)

300

f(x) = √(4x³+5)

f'(x) = 6x² / √(4x³+5)

400

f(x) = sin((3x-1)²)

f'(x) = 6 * (3x-1) * cos((3x-1)²)

500

f(x) = 3tan(√x)

f'(x) =3/2√x * sec²(√x)

Product Rule

100

f(x) = (7x³ + 1) * (9x² + 8)

f'(x) = (7x³ + 1) * (18x) + (9x² + 8) * (21x²)

200

f(x) = x² * sin(x)

f'(x) = 2x*sinx+ x²*cosx

300

f(x) = sin(2x)*cos(3x)

f'(x) = 2cos(2x)cos(3x)−3sin(2x)sin(3x)

400

f(x) = x² * tan(4x) 

f'(x) = 2x*tan(4x) + 4x²*sec²4x

500

f(x) = (x² + 3x) * sin(√x) 

f'(x) = (2x+3)sin(√x) + [ (x² + 3x)*cos(√x) / (2√x) ]

Quotient Rule

100

f(x)= (x²-1)³/ x²+1

f′(x)= [4x(x²-1)²(x²+2)] / (x²+1)²

200

f(x) = x / √(x+1)  

f'(x) = (x+2) / 2(x+1)^3/2

300

f(x) = sin(x) / x

f'(x) = (x*cos(x) - sin (x))/x²

400

f(x) = tan(x) / x² + 1

f'(x) = [(x²+1)(sec²x) − (2x)(tanx)] / (x²+1)²

500

f(x) = sec(x) / x²

f'(x) = [sec(x) * (x²tanx−2x)] / x⁴

Trigonometric

100

f(x) = cosx

f'(x) = -sinx

200

f(x) = sin(2x)

f'(x) = 2cos(2x)

300

f(x) = cos²(2x)

f'(x) = −4cos(2x)sin(2x)= −2sin(4x) 

400

f(x) = arcsec(x)

f'(x) = 1 / |x| √(x² - 1)

500

f(x) = sec(x²-x)

f'(x) = (2x−1) * sec(x²−x)* tan(x²−x) 

Hard (Bonus)

100

f(x) = [ (1-sin(x)) / cos(x)]

f'(x) = (sin(x) - 1) * sec²(x) 

200

f(x) = (sin(x)cos(x)) / (1/2*sin(x))

f'(x) = -2sinx

300

f(x) = sin[(x-1)/(x+1)]

f'(x) = 2/(x+1)² * cos[(x-1)/(x+1)]

400

f(x) = sin(x) / √(1+x²)

f'(x) = [cos(x) / √(1+x²)] -  [x*sin(x) / (1+x²)^3/2]

f'(x) = [cos(x)*(1+x²) -  (x*sin(x))] / (1+x²)^3/2

500

f(x) = cos(x²) * √(x²+1) 

f'(x) = −2xsin(x²)*√(x2+1) + [x*cos(x²) / √(x²+1)]