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Differential Calculus

Chapter 2 · Differentiation Rules

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Power Rule
ddx[xn]=n x n−1\frac{d}{dx}\left[x^n\right] = n\,x^{\,n-1}
Any real nn.
Constant multiple
ddx[c f]=c f′\frac{d}{dx}[c\,f] = c\,f'
Sum / difference
ddx[f±g]=f′±g′\frac{d}{dx}[f \pm g] = f' \pm g'
Chain Rule
dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}
Derivative of outer (at inner) × derivative of inner.
Product Rule
(fg)′=f′g+fg′(fg)' = f'g + fg'
Quotient Rule
(fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}
Common derivatives
ddx[ex]=ex,    ddx[ln⁡x]=1x,    ddx[sin⁡x]=cos⁡x,    ddx[cos⁡x]=−sin⁡x\frac{d}{dx}[e^x]=e^x,\;\; \frac{d}{dx}[\ln x]=\tfrac{1}{x},\;\; \frac{d}{dx}[\sin x]=\cos x,\;\; \frac{d}{dx}[\cos x]=-\sin x
Product / quotient
(uv)′=u′v+uv′,    (uv)′=u′v−uv′v2(uv)' = u'v + uv',\;\; \left(\tfrac{u}{v}\right)' = \tfrac{u'v - uv'}{v^2}
Chain rule
dydx=dydu⋅dudx\tfrac{dy}{dx} = \tfrac{dy}{du}\cdot\tfrac{du}{dx}
Key derivatives
(sin⁡x)′=cos⁡x,  (ex)′=ex,  (ln⁡x)′=1x(\sin x)'=\cos x,\;(e^x)'=e^x,\;(\ln x)'=\tfrac1x
Common trap
Trig derivatives assume x in RADIANS; don’t forget the chain rule’s inner derivative.