Differentiation

Standard derivatives:

aa

00


xax^a

axa−1ax^{a-1}


exe^{x}

exe^x


eaxe^{ax}

aeaxae^{ax}


axa^x

axln(a)a^{x}ln\left(a\right)

if a>0a>0

ln(x)ln(x)

1x\frac{1}{x}

if x>0x>0

log(x)log(x)

1x⋅ln⁡(10)\frac{1}{x\cdot\ln\left(10\right)}

if x>0x>0

sin(x)sin(x)

cos(x)cos(x)


cos(x)cos(x)

−sin(x)-sin(x)


tan(x)tan(x)

1cos⁡(x)2\frac{1}{\cos\left(x\right)^2}



Quotient rule for [f(x)g(x)]′[\frac{f\left(x\right)}{g\left(x\right)}]^{\prime}

→ uv\frac{u}{v}

→ f(x)’g(x)−f(x)g(x)’g(x)2\frac{f(x)’g(x)-f(x)g(x)’}{g\left(x\right)^2} → u’v−uv’v2\frac{u’v-uv’}{v^2}

  1. Find derivative for both numerator (u’) and denumerator (v’)

  2. Derivative numerator times denumerator (u′⋅vu^{\prime}\cdot v ), minus derivative denumerator times numerator (u⋅v′u\cdot v^{\prime}).

  3. Divided by denumerator squared (v2v²)

e.g.: 2x−34x2\frac{2x-3}{4x^2}

  1. [2x−3]’=2[2x - 3]’ = 2, [4x2]’=8x[4x^2]’=8x

  2. 2⋅4x2−8x(2x−3)2 \cdot 4x² - 8x(2x-3) = 8x2−16x2+24x8x² - 16x² + 24x = −8x2+24x-8x^{2}+24x

  3. −8x2+24x16x4\frac{-8x^2+24x}{16x^4} = −x+32x3\frac{-x+3}{2x^3} = 3−x2x3\frac{3-x}{2x^3}


Product rule for [f(x)⋅g(x)]′[f\left(x\right)\cdot g\left(x\right)]^{\prime}

→ u⋅vu \cdot v

→ f’(x)⋅g(x)+f(x)⋅g’(x)f’(x) \cdot g(x) + f(x) \cdot g’(x) → u’v+uv’u’v + uv’

  1. Find derivative for both sides (u’ and v’)

  2. Each side of the “⋅\cdot” becomes its own derivative times the other side (u’⋅vu’ \cdot v and v’⋅uv’\cdot u )

  3. Connect sides by addition

e.g. (3x−1)⋅x(3x-1)\cdot\sqrt{x}

  1. [3x−1]’=3[3x-1]’=3, [x]′=12x−12=12x[\sqrt{x}]^{\prime}=\frac12x^{-\frac12}=\frac{1}{2\sqrt{x}}

  2. 3⋅x3\cdot\sqrt{x} + 12x⋅(3x−1)\frac{1}{2\sqrt{x}}\cdot(3x-1) = 3x+3x−12x3\sqrt{x}+\frac{3x-1}{2\sqrt{x}}

2,5. 3x=1⋅3x=2x2x⋅3x1=2x12⋅3x122x⋅1=2⋅3⋅x12+122x=6x2x3\sqrt{x}=1\cdot3\sqrt{x}=\frac{2\sqrt{x}}{2\sqrt{x}}\cdot\frac{3\sqrt{x}}{1}=\frac{2x^{\frac12}\cdot3x^{\frac12}}{2\sqrt{x\cdot1}}=\frac{2\cdot3\cdot x^{\frac12+\frac12}}{2\sqrt{x}}=\frac{6x}{2\sqrt{x}}

  1. 6x2x+3x−12x\frac{6x}{2\sqrt{x}}+\frac{3x-1}{2\sqrt{x}} = 9x−12x\frac{9x-1}{2\sqrt{x}}


Chain rule for composite functions [f(x)=g(u(x))]’[f(x) = g(u(x))]’

→ g’(u)⋅u’g’(u) \cdot u’

  1. Identify g(u) and u

  2. Find derivative of both

  3. Combine

e.g. ln(x)3ln(x)³

  1. u=ln(x)u = ln(x), g(u)=u3g(u) = u³

  2. [ln(x)]’=1x[ln(x)]’=\frac{1}{x} , [u3]’=3u2[u³]’= 3u²

  3. 3ln(x)2⋅1x3ln(x)^2\cdot\frac{1}{x} = 3ln(x)2x\frac{3ln(x)²}{x}