# Difference between revisions of "Derivative"

The derivative of a function is defined as the instantaneous rate of change of the function with respect to one of the variables. Note that not every function has a derivative.

## Notation

The derivative of f(x) can be expressed in several ways including:

• $\frac{d}{dx}$
• $f'(x)$
• $f'$

## Finding the Derivative

For any constant, the derivative is 0.

For any monomial $nx$, the derivative is n.

Note that when we take the derivative of any polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$, we can take the derivative of each addend and then add these to find the derivative of the polynomial.

The chain rule states that the derivative of any $ax^n$ is $anx^{n-1}$

To find the derivative of $f(x) \cdot g(x)$ we cannot do what we did with addition. We must instead use the product rule: $(f(x) \cdot g(x))' = f'g + g'f$

The quotient rule states that $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$

The following pages provide additional information on derivatives.