calc 3

\(\frac{d}{dx}[\ln u]\)

\(\frac{u'}{u}\)

\(\frac{d}{dx}[\text{arccot }x]\)

\(-\frac{1}{1+x^2}\)

\(x'  \text{in numerator for chain rule}\)

\(\frac{d}{dx}[\arctan x]\)

\(\frac{1}{1+x^2}\)

\(x'  \text{in numerator for chain rule}\)

Chain rule short cut

\(\frac{dy}{dx}n[u(x)]^{n-1}\cdot \frac{du}{dx}\)

\(\frac{dy}{dx}[u^n]=nu^{n-1}\cdot u'\)

\(\frac{d}{dx}[e^u]\)

\(e^u\cdot u'\)

\(\frac{d}{dx}[\log_a u]\)

\(\frac{1}{(\ln a)u}\cdot \frac{du}{dx}\)

Trig functions chain rule

\(\frac{d}{dx}[\text{trig }u]=(\text{trig }u)'\cdot u'\)

\(s(t)\)

Position function

\(x^{-1}\)

\(f^{-1}(x)\)

\(\frac{1}{x}\)

inverse of \(f(x)\)

\(\frac{d}{dx}[\arcsin x]\)

\(\frac{1}{\sqrt{1-x^2}}\)

\(x'  \text{in numerator for chain rule}\)

\(v(t)=s'(t)\)

Velocity function

\(\frac{d}{dx}[\arccos x]\)

\(-\frac{1}{\sqrt{1-x^2}}\)

\(x'  \text{in numerator for chain rule}\)

\(\frac{d}{dx}[\text{arccsc }x]\)

\(-\frac{1}{|x|\sqrt{x^2-1}}\)

\(x'  \text{in numerator for chain rule}\)

Chain rule

\(\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)\)

\(\frac{d}{dx}[\text{arcsec } x]\)

\(\frac{1}{|x|\sqrt{x^2-1}}\)

\(x'  \text{in numerator for chain rule}\)

\(\frac{d}{dx}[a^u]\)

\((\ln a)a^u\cdot \frac{du}{dx}\)

\(a(t)=v'(t)=s''(t)\)

Acceleration function

\(\frac{d}{dx}[f^{-1}(x)]\)

\(\frac{1}{f'(f^{-1}(x))}\)