Assume that \(c\) and \(n\) are real numbers and \(f(x), g(x),\) and \(u(x)\) are any differentiable functions of \(x\):
- \(\frac{d}{dx}(c)=0\)
- \(\frac{d}{dx}(x^n)=nx^{n-1}\) (power rule)
- \(\frac{d}{dx}(cf)=c\frac{df}{dx}\)
- \(\frac{d}{dx}(f\pm g)=\frac{df}{dx}\pm\frac{dg}{dx}\)
- \(\frac{d}{dx}(fg)=g\frac{df}{dx}+f\frac{dg}{dx}\) or \((fg)^{\prime}=gf^{\prime}+fg^{\prime}\) (product rule)
- \(\frac{d(f/g)}{dx}=\frac{g\frac{df}{dx}-f\frac{dg}{dx}}{g^2}\) or \((fg)^{\prime}=gf^{\prime}+fg^{\prime}\) (quotient rule)
- \(\frac{d[u(x)]^n}{dx}=n[u(x)]^{n-1}\frac{d(u(x))}{dx}\) (general power rule - chain rule)
- \(\frac{d(e^x)}{dx}=e^x\)
- \(\frac{d(\ln x)}{dx}=\frac1x\)
- \(\frac{d(e^{u(x)})}{dx}=e^{u(x)}\frac{d(u(x))}{dx}\)
- \(\quad\frac{d[\ln(u(x))]}{dx}=\frac1{u(x)}\frac{d(u(x))}{dx}\)
- \(\frac{d(a^x)}{dx}=a^x\ln a\)
- \(\frac{d(\log_ax)}{dx}=\frac1{x\ln a}\)
- \(\frac{d(a^{u(x)})}{dx}=a^{u(x)}\ln a\frac{d(u(x))}{dx}\)
- \(\frac{d(\log_au(x))}{dx}=\frac{1}{u(x)\ln a}\frac{d(u(x))}{dx}\)