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DERIVATIVE FORMULAS

DERIVATIVE FORMULAS

DERIVATIVE FORMULAS

Derivative Formulas

Constant Rule

ddx[c]=0\frac{d}{dx} [c] = 0 

Basic Rule

ddx[x]=1\frac{d}{dx} [x] = 1 

Sum Rule

ddx[u+v]=u′+v′\frac{d}{dx} [u + v] = u’ + v’ 

Difference Rule

ddx[u−v]=u′−v′\frac{d}{dx} [u – v] = u’ – v’ 

Product Rule

ddx[uv]=u′v+uv′\frac{d}{dx} [uv] = u’v + uv’

Quotient Rule

ddx[uv]=u′v−uv′v2\frac{d}{dx} \left[\frac{u}{v}\right] = \frac{u’v – uv’}{v^2}

Without Chain Rule

Power Rule

  • ddx[xn]=nxn−1\frac{d}{dx} [x^n] = nx^{n-1} 
  • ddx[cxn]=cnxn−1\frac{d}{dx} [cx^n] = cnx^{n-1} 

Exponential

  • ddx[ex]=ex\frac{d}{dx} [e^x] = e^x 

Natural Log

  • ddx[ln⁡x]=1x\frac{d}{dx} [\ln x] = \frac{1}{x}

Logarithmic

  • ddx[log⁡ax]=1ln⁡a⋅1x\frac{d}{dx} [\log_a x] = \frac{1}{\ln a} \cdot \frac{1}{x}

Absolute Value

  • ddx[∣x∣]=x∣x∣\frac{d}{dx} [|x|] = \frac{x}{|x|}

Trig Functions

  • ddx[sin⁡x]=cos⁡x\frac{d}{dx} [\sin x] = \cos x 
  • ddx[cos⁡x]=−sin⁡x\frac{d}{dx} [\cos x] = -\sin x 
  • ddx[tan⁡x]=sec⁡2x\frac{d}{dx} [\tan x] = \sec^2 x 
  • ddx[csc⁡x]=−csc⁡xcot⁡x\frac{d}{dx} [\csc x] = -\csc x \cot x 
  • ddx[sec⁡x]=sec⁡xtan⁡x\frac{d}{dx} [\sec x] = \sec x \tan x 
  • ddx[cot⁡x]=−csc⁡2x\frac{d}{dx} [\cot x] = -\csc^2 x 

Inverse Trig Functions

  • ddx[sin⁡−1x]=11−x2\frac{d}{dx} [\sin^{-1} x] = \frac{1}{\sqrt{1 – x^2}}
  • ddx[cos⁡−1x]=−11−x2\frac{d}{dx} [\cos^{-1} x] = -\frac{1}{\sqrt{1 – x^2}}
  • ddx[tan⁡−1x]=11+x2\frac{d}{dx} [\tan^{-1} x] = \frac{1}{1 + x^2}
  • ddx[csc⁡−1x]=−1∣x∣x2−1\frac{d}{dx} [\csc^{-1} x] = -\frac{1}{|x| \sqrt{x^2 – 1}}
  • ddx[sec⁡−1x]=1∣x∣x2−1\frac{d}{dx} [\sec^{-1} x] = \frac{1}{|x| \sqrt{x^2 – 1}}
  • ddx[cot⁡−1x]=−11+x2\frac{d}{dx} [\cot^{-1} x] = -\frac{1}{1 + x^2}

With Chain Rule

Chain Rule

  • ddx[un]=nun−1u′\frac{d}{dx} [u^n] = nu^{n-1} u’ 
  • ddx[cun]=cnun−1u′\frac{d}{dx} [cu^n] = cnu^{n-1} u’ 

Exponential

  • ddx[eu]=euu′\frac{d}{dx} [e^u] = e^u u’ 

Natural Log

  • ddx[ln⁡u]=1uu′\frac{d}{dx} [\ln u] = \frac{1}{u} u’ 

Logarithmic

  • ddx[log⁡au]=1ln⁡a⋅1u⋅u′\frac{d}{dx} [\log_a u] = \frac{1}{\ln a} \cdot \frac{1}{u} \cdot u’ 

Absolute Value

  • ddx[∣u∣]=u∣u∣u′\frac{d}{dx} [|u|] = \frac{u}{|u|} u’ 

Trig Functions

  • ddx[sin⁡u]=(cos⁡u)u′\frac{d}{dx} [\sin u] = (\cos u) u’ 
  • ddx[cos⁡u]=−(sin⁡u)u′\frac{d}{dx} [\cos u] = -(\sin u) u’ 
  • ddx[tan⁡u]=(sec⁡2u)u′\frac{d}{dx} [\tan u] = (\sec^2 u) u’ 
  • ddx[csc⁡u]=−(csc⁡ucot⁡u)u′\frac{d}{dx} [\csc u] = -(\csc u \cot u) u’ 
  • ddx[sec⁡u]=(sec⁡utan⁡u)u′\frac{d}{dx} [\sec u] = (\sec u \tan u) u’ 
  • ddx[cot⁡u]=−(csc⁡2u)u′\frac{d}{dx} [\cot u] = -(\csc^2 u) u’ 

Inverse Trig Functions

  • ddx[sin⁡−1u]=u′1−u2\frac{d}{dx} [\sin^{-1} u] = \frac{u’}{\sqrt{1 – u^2}}
  • ddx[cos⁡−1u]=−u′1−u2\frac{d}{dx} [\cos^{-1} u] = -\frac{u’}{\sqrt{1 – u^2}}
  • ddx[tan⁡−1u]=u′1+u2\frac{d}{dx} [\tan^{-1} u] = \frac{u’}{1 + u^2}
  • ddx[csc⁡−1u]=−u′∣u∣u2−1\frac{d}{dx} [\csc^{-1} u] = -\frac{u’}{|u| \sqrt{u^2 – 1}}
  • ddx[sec⁡−1u]=u′∣u∣u2−1\frac{d}{dx} [\sec^{-1} u] = \frac{u’}{|u| \sqrt{u^2 – 1}}
  • ddx[cot⁡−1u]=−u′1+u2\frac{d}{dx} [\cot^{-1} u] = -\frac{u’}{1 + u^2}

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March 11, 2025 No Comments