Math 19A

Product Rule

100

Differentiate f(x) = xe^x.

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

200

Differentiate g(x) = x^2 \sin x

g(x) = 2x sin x + x^2 cos x

300

Find (d/{d}{dx})(x ln x).

( d/{d}{dx})(x ln x) = ln x + 1

400

Differentiate h(x) = (3x^2)(cos x).

h(x) = 6x cos x - 3x^2 sin x

500

Differentiate y = (x^2 + 1)e^{x^3}.

y = 2x e^{x^3} + (x^2 + 1)(3x^2 e^{x^3}) = e^{x^3} (2x + 3x^2(x^2+1))

Quotient Rule

100

Differentiate f(x) = x/(x+1)

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

200

Differentiate g(x) = sin x/x

g(x) = (x cos x — sin x)/(x^2)

300

Find d/dx (x^3/cos x)

d/dx (x^3 sec x) = 3x^2 sec x + x^3 sec x tan x 

400

Differentiate h(x) = e^x/x^2

h(x) = (e^x x^2 — e^x times 2x) / (x^4) = (e^x (x^2 — 2x) / (x^4) = e^x (x-2) / x^3

500

Differentiate y = (x^2 + 1) / sqrt x

y = (3x^2 - 1) / 2x^(3/2)

Chain Rule

100

Differentiate f(x) = (x^2 + 1)^3

f(x) = 3(x^2 +1)^2 times 2x = 6x (x^2 +1)^2

200

Find d/dx (sin(5x))

= 5 cos (5x)

300

Differentiate g(x) = e^2x{^3}

g(x) = e^2x{3} times 6x^2 = 6x^2 e^2x{^3}

400

Find d/dx (ln(4x^2 + 3))

(8x)/(4x^2 + 3)

500

Differentiate y = cos (x^2 sin x)

y = — sin (x^2 sin x) (2x sin x +x^2 cos x)

Logarithmic Differentiation

100

Differentiate y = x^x using log diff

y = x^x (ln x + 1)

200

Differentiate y= (x^2 + 3x)^5 (logs optional)

y= 5(x^2 + 3x)^4 (2x + 3)

300

Y = (x^3(x+1)^2) / sqrt x — 1. Use log diff.

y/y = 3/x + 2/x+1 — 1/2(x-1) so 

y = (x^3(x+1)^2) / sqrt x-1 ((3/x) + (2/x+1) — (1/2(x-1))

400

Y = (x^2 + 1)^ln x

y = (x^2 +1)^ln x ((ln(x^2 +1)/x) + (2x ln x) / (x^2 + 1))

500

Y = x^x sin^x (x^2) times (3x+1)^(1/x)

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Trigonometric Differentiation

100

Differentiate f(x) = sin x

f(x) = cos x 

200

Differentiate g(x) = cos x

g(x) = — sin x

300

Differentiate h(x) = tan x 

h(x) = sec^2 x

400

Find d/dx (x sec x)

sec x + x sec x tan x

500

Differentiate y = sin (x^2) cos (x^2)

y = 2x cos(x^2) cos(x^2) — 2x sin (x^2) sin (x^2) or y = 2x cos(2x^2)