Derivatives and their Rules

Power Rule

100

The derivative of

-4x^4

-16x^3

200

3/2x^6 + 4/5 x^10

9x^5 + 8x^9

300

100x^-2 + 7x - 1

-200x^-3 + 7

400

-3x^-4 + x^{3/2} -90

12x^-5 + 3/2 x^{1/2}

500

e^3x^{-4/3} + (x-1)^-2 - x^7

(-4/3)e^3x^{-7/3} - 2(x-1)^{-3} - 7x^6

Product Rule

100

The derivative of

x^3(x+1)

3x^2(x+1) + x^3

200

e^x x^3

e^x x^3 + e^x (3x^2)

300

ln(x)/x

1/x(1/x) + ln(x)(-1/x^2)

400

sin(x)cos(x)

cos^2(x) - sin^2(x)

500

x^5e^xtan(x)

x^5 e^x sin(x) +e^x(x^5 cosx + 5x^4 sin x)

Quotient Rule

100

The derivative of

{x+1}/x

-1/x^2

200

x^3/{3x^2-x}

{(3x^2-x) 3x^2 - x^3 (6x - 1)}/{(3x^2 - x)^2}

300

{e^x + x}/{ln(x)}

{ln(x) (e^x + 1) - (e^x + x)1/x}/{(ln(x))^2}

400

{x^{-2} - sinx}/{cosx}

{cos x (-2x^-3 - cos x) - (x^{-2} - sin x)(-sin x)}/{cos^2 x}

500

{1/x + cos x}/{e^x}

- {e^-x(x^2 sin x + x^2 cos x + x + 1)}/{x^2}

Chain Rule

100

The derivative of

(3x^2 + x)^5

5(3x^2 + x)^4 (6x + 1)

200

e^{x^{2/3}}

e^{x^{2/3}} (2/3x^{-1/3})

300

sin(ln(x))

cos(ln(x))/x

400

(2/3x^{1/5} + 1)^{-2}

-2(2/3x^{1/5} + 1)^{-3} (2/15 x^{-4/5})

500

ln(cos(e^{x^3 - x^2}))

1/cos(e^{x^3 - x^2}) (-sin(e^{x^3 - x^2})) e^{x^3 - x^2} (3x^2 - 2x)

Grab bag

100

The derivative of

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0

200

2/3(x^2 - 5x + 6)^{3/2}

(x^2 - 5x + 6)^{1/2}(2x - 5)

300

2/{x^2-1}

-{4x}/{(x^2-1)^2}

400

sec(x + 1)

tan(x+1)sec(x+1)

500

sec(x) tan(x)

sec(x) tan^2(x) + sec^3(x)