Calculus Derivative Guide

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36 Terms

1
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Derivative of sin x

cos x

<p><sup>cos x</sup></p>
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Derivative of cos x

-sin x

<p>-sin x</p>
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Derivative of tan x

sec2 x

<p>sec<sup>2 </sup>x</p>
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Derivative of cot x

-csc2 x

<p>-csc<sup>2 </sup>x</p>
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Derivative of sec x

sec x tan x

<p>sec x tan x</p>
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Derivative of csc x

-cot x csc x

<p>-cot x csc x</p>
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Derivative of ex

ex

<p>e<sup>x</sup></p>
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Product Rule

f(x)*g’(x) + g(x)*f’(x)

<p><span>f(x)*g’(x) + g(x)*f’(x)</span></p>
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Quotient Rule f(x) / g(x)

(g(x)*f’(x) - f(x)*g’(x)) / (g(x))2

<p><span>(g(x)*f’(x) - f(x)*g’(x)) / (g(x))</span><sup>2</sup></p>
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Power Rule (derivative of xn)

nxn-1

  • xÂł=3x²

<p>nx<sup>n-1</sup></p><ul><li><p>x³=3x²</p></li></ul><p></p>
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Constant Rule (derivative of a constant)


0

  • 5=0

  • pi=0

<p><br>0</p><ul><li><p>5=0</p></li><li><p>pi=0</p></li></ul><p></p>
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Constant Multiple Rule (cf(x))

cf’(x)

  • 5x4=20xÂł

<p>cf’(x)</p><ul><li><p>5x<sup>4</sup>=20x³</p></li></ul><p></p>
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Sum and Difference Rules (derivative of f(x) ± g(x))


f’(x) ± g’(x)

  • x4+3x2-1 = 4x3+6x

<p><br>f’(x) ± g’(x)</p><ul><li><p>x<sup>4</sup>+3x<sup>2</sup>-1 = 4x<sup>3</sup>+6x</p></li></ul><p></p>
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Derivative of sin-1 x

1 / sqrt(1 - x2)

<p>1 / sqrt(1 - x<sup>2</sup>)</p>
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Derivative of cos-1 x

-1 / sqrt(1 - x2)

<p>-1 / sqrt(1 - x<sup>2</sup>)</p>
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Derivative of tan-1 x

1 / 1 + x2

<p>1 / 1 + x<sup>2</sup></p>
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Derivative of cot-1 x

-1 / 1 + x2

<p>-1 / 1 + x<sup>2</sup></p>
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Derivative of sec-1 x

1 / |x|*sqrt(x2 - 1)

<p>1 / |x|*sqrt(x<sup>2</sup> - 1)</p>
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Derivative of csc-1 x

-1 / |x|*sqrt(x2 - 1)

<p>-1 / |x|*sqrt(x<sup>2</sup> - 1)</p>
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Chain Rule f(g(x))

f’(g(x))*g’(x)

<p>f’(g(x))*g’(x)</p>
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Derivative of ln x or loge x

1 / x

<p>1 / x</p>
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Derivative of loga x

1 / x*ln n

<p>1 / x*ln n</p>
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Implicit Rule (dy/dx (x2+y2=25)

2x + 2y*y’ = 0

y’ = -x / y

<p>2x + 2y*y’ = 0</p><p>y’ = -x / y</p>
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Expanding Rule (3Ă—2 + 1)2

(3Ă—2 + 1)*(3Ă—2 + 1)

(9Ă—4 + 6Ă—2 + 1)

y’ = 36×3 + 12x

<p>(3×<sup>2</sup> + 1)*(3×<sup>2</sup> + 1)</p><p>(9×<sup>4</sup> + 6×<sup>2</sup> + 1)</p><p>y’ = 36×<sup>3</sup> + 12x</p>
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Outside-Inside Rule (3Ă—2 + 1)2

(3Ă—2 + 1)2

y’ = 2(3×2 + 1)*6x

y’ = 36×3 + 12x

<p>(3×<sup>2</sup> + 1)<sup>2</sup></p><p>y’ = 2(3×<sup>2</sup> + 1)*6x</p><p>y’ = 36×<sup>3</sup> + 12x</p>
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U - Substitution (3Ă—2 + 1)2

(3Ă—2 + 1)2

y’ = 2u * 6x

y’ = 2(3×2 + 1)2 * 6x

y’ = 36×3 + 12x

<p>(3×<sup>2</sup> + 1)<sup>2</sup></p><p>y’ = 2u * 6x</p><p>y’ = 2(3×<sup>2</sup> + 1)<sup>2 </sup> * 6x</p><p>y’ = 36×<sup>3</sup> + 12x</p>
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|x| is equivalent to

Sqrt(x²)

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Derivative of ax

ax * lna

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Derivaticve of lnex or elnx

x

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y = logb x is equivilant to

by = x

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y = log10 x is equivilant to

y = log x

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y = loge x is equivilant to

y = ln x

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When logb(xy)

logb x + logb y or ln x + ln y becauce bm * bn = bm+n

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When logb(x/y)

logb x - logb y or ln x - ln y becauce bm/bn = bm-n

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When logb xp

P * logb x or P * ln x becauce (bm)n = bm*n

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Derivative lny = any function

y*(derivative of that function

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