When we wish to differentiate complicated expressions, a possible way to differentiate the expression is to expand it and get a polynomial, and then differentiate that polynomial. This method becomes very complicated and is particularly error prone when doing calculations by hand. In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided by using differentiation rules. Some of the most basic rules are the following.

The Constant Rule

If $f(x)$ is a constant, then $f'(x) = 0$, since the rate of change of a constant is always zero.

The Sum Rule

$(\alpha f + \beta g)' = \alpha f' + \beta g'$

for all functions $f$ and $g$ and all real numbers $\alpha$ and $\beta$.

The Product Rule

$(fg)' = f'g + g'f$

for all functions $f$ and $g$.

By extension, this means that the derivative of a constant times a function is the constant times the derivative of the function.

The Quotient Rule

$\displaystyle{\left ( \frac {f}{g} \right )' = \frac {f'g - fg'}{g^2}}$

for all functions $f$ and $g$ at all inputs where $g \neq 0$.

Model Rockets

The flight of model rockets can be modeled using the product rule.

The Chain Rule

If $f(x) = h(g(x))$, then $f'(x) = h'(g(x)) g'(x)$.

Example

Consider the following function:

$f(x) = x^4 + e^{x^2} - ln(x)e^x + 7$

Differentiating yields:

$\displaystyle{f'(x) = 4x^{(4-1)} + \frac{d(x^2)}{dx}e^{x^2} - \frac {d(ln\:x)}{dx}e^x - ln\:x\frac{d(e^x)}{dx} + 0 \\ \qquad = 4x^3 + 2xe^{x^2} - \frac {1}{x}e^x - ln(x)e^x}$

Here the second term was computed using the chain rule and the third using the product rule. The known derivatives of the elementary functions $x^2$, $x^4$, $\ln(x)$, and $e^x$, as well as that of the constant 7, were also used.