With Chain Rule

Chain Rule

• $\frac{d}{dx} [u^n] = nu^{n-1} u’$ 
• $\frac{d}{dx} [cu^n] = cnu^{n-1} u’$ 

Exponential

• $\frac{d}{dx} [e^u] = e^u u’$ 

Natural Log

• $\frac{d}{dx} [\ln u] = \frac{1}{u} u’$ 

Logarithmic

• $\frac{d}{dx} [\log_a u] = \frac{1}{\ln a} \cdot \frac{1}{u} \cdot u’$ 

Absolute Value

• $\frac{d}{dx} [|u|] = \frac{u}{|u|} u’$ 

Trig Functions

• $\frac{d}{dx} [\sin u] = (\cos u) u’$ 
• $\frac{d}{dx} [\cos u] = -(\sin u) u’$ 
• $\frac{d}{dx} [\tan u] = (\sec^2 u) u’$ 
• $\frac{d}{dx} [\csc u] = -(\csc u \cot u) u’$ 
• $\frac{d}{dx} [\sec u] = (\sec u \tan u) u’$ 
• $\frac{d}{dx} [\cot u] = -(\csc^2 u) u’$ 

Inverse Trig Functions

• $\frac{d}{dx} [\sin^{-1} u] = \frac{u’}{\sqrt{1 – u^2}}$
• $\frac{d}{dx} [\cos^{-1} u] = -\frac{u’}{\sqrt{1 – u^2}}$
• $\frac{d}{dx} [\tan^{-1} u] = \frac{u’}{1 + u^2}$
• $\frac{d}{dx} [\csc^{-1} u] = -\frac{u’}{|u| \sqrt{u^2 – 1}}$
• $\frac{d}{dx} [\sec^{-1} u] = \frac{u’}{|u| \sqrt{u^2 – 1}}$
• $\frac{d}{dx} [\cot^{-1} u] = -\frac{u’}{1 + u^2}$