MATH 191 Derivative Review

Product Rule

100

y = (2x+1)(x²+1)

What is dy/dx = (2x+1)(2x)+(2)(x²+1)

200

y=sqrtx tan(x)

y'=sqrtx sec^2(x)+tan(x)/{2sqrtx}

300

y = x^2 sin(x)e^x

dy/dx=2xsin(x)e^x+x^2 cos(x)e^x+x^2sin(x)e^x

400

y = sec(x)(x⁴+3x²)(x+3)

What is dy/dx = sec(x)[(x⁴+3x²)+(x+3)(4x³+6x)]+sec(x)tan(x)(x⁴+3x²)(x+3)

500

y=x sin(x)cos(x)ln(x)

y'=sin(x)cos(x)ln(x)+xcos^2(x)ln(x)-xsin^2(x)ln(x)+sin(x)cos(x)

Quotient Rule

100

y = x²/(x+3)

What is dy/dx = [(x+3)(2x)-(x²)]/(x+3)²

200

y = sin(x)/x

What is dy/dx = [xcos(x)-sin(x)]/x²

300

y = tan(x)/e^x

What is dy/dx = [e^xsec²(x)-tan(x)e^x]/e^2x

400

y = e^2x(x²+1)/(x-3)

What is dy/dx = [(x-3)(e^2x(2x)+2e^2x(x²+1))-e^2x(x²+1)]/(x-3)²

500

y = 2x/[e^xcot(x)]

What is dy/dx = [2e^xcot(x) - 2x(e^x(-csc²(x))+e^xcot(x)]/[e^xcot(x)]²

Chain Rule

100

y = (3x²+2x-1)⁹

What is dy/dx = 9(3x²+2x-1)⁸(6x+2)

200

y = cos⁴(x)

What is dy/dx = -4cos³(x)sin(x)

300

y = sec(3e^x)

What is dy/dx = sec(3e^x)tan(3e^x)(3e^x)

400

y = sin(e^x2)

What is dy/dx = cos(e^x2) e^x2(2x)

500

y = sqrt(cos²(x))e^2x

What is dy/dx = 2sqrt(cos²(x))e^2x+e^2xcos(x)sin(x)/sqrt(cos²)

Implicit Differentiation

100

2x²+y³ = 3

What is dy/dx = -4x/3y²

200

x²y = 10x

What is dy/dx = (10-2xy)/x²

300

cos(xy) = y²

What is dy/dx = (2y + sin(xy))/(-ysin(xy))

400

cot(x/y)= y

What is dy/dx = (ycsc²(x/y))/[xcsc²(x/y)-y²]

500

(x²+y²)² = cos(e^y)

What is dy/dx = [-4x(x²+y²]/[4y(x²+y²)+sin(e^y)e^y]

Inverse Trig and Log

100

y=ln(x)

dy/dx=1/x

200

y=log_2(x)-x^2

dy/dx=1/{log(2)x}-2x

300

y=arctan(x)+arcsin(x)

dy/dx=1/{1+x^2}+1/\sqrt{1-x^2}

400

y=sec^{-1}(ln(x))

dy/dx=1/{xln(x)\sqrt{ln(x)^2-1}}

500

y=x^sin(x)

dy/dx=(sin(x)/x+cos(x)ln(x))x^sin(x)