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:

$\frac{d}{dx}$
$\displaystyle f'(x)$
$\displaystyle f'$

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 $\displaystyle a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ , 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 $\displaystyle ax^n$ is $\displaystyle anx^{n-1}$

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

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

The following pages provide additional information on derivatives.

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Derivative

Notation

Finding the Derivative

See also