Elementary Differentiation

The following differentiation rules may used at some point during the course. Note that $c$ is a constant. We will not require you to differentiate trigonometric or hyperbolic functions.

General Formulas

$$ \begin{align} &\frac{\mathrm{d}}{\mathrm{d}x} c= 0\\ &\frac{\mathrm{d}}{\mathrm{d}x}\big[f(x)+g(x)\big] = f^{\prime}(x)-g^{\prime}(x)\\ &\frac{\mathrm{d}}{\mathrm{d}x}\big[g(x)f(x)\big] = f(x)g^{\prime}(x)+g(x)f^{\prime}(x)\\ &\frac{\mathrm{d}}{\mathrm{d}x}f(g(x))=f^{\prime}(g(x))g^{\prime}(x)\\ &\frac{\mathrm{d}}{\mathrm{d}x}\big[c f(x)\big] =c f^{\prime}(x)\\ &\frac{\mathrm{d}}{\mathrm{d}x}\Big[\frac{f(x)}{g(x)}\Big] = \frac{g(x)f^{\prime}(x)-f(x)f^{\prime}(x)}{\big[g(x)\big]^2}\\ &\frac{\mathrm{d}}{\mathrm{d}x}x^n=nx^{n-1} \end{align} $$

Exponents and Logarithmic Functions

$$ \begin{align} & \frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x\\ &\frac{\mathrm{d}}{\mathrm{d}x} a^x = a^x\ln{a}\\ &\frac{\mathrm{d}}{\mathrm{d}x} \ln{|x|} = \frac{1}{x}\\ &\frac{\mathrm{d}}{\mathrm{d}x} \log_a{x} = \frac{1}{x \ln{a}} \end{align} $$