This page reviews the following aspect of differentiation:

- Definition of the Derivative
- The Product Rule
- The Quotient Rule
- The Chain Rule
- Examples involving Partial Derivatives

**The Definition of the Derivative**

Although you will likely not use the definition to compute derivatives in later calculus courses, you should nevertheless know the definition of the derivative. The derivative of f with respect to x is given by

when this limit exists.

If f(x) and g(x) are differentiable functions, then the derivative of the product fg with respect to x is given by

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

This can be fairly easily derived from the definition.

**Example**

To calculate the derivative of h(x) = x^{2}e^{x} we must
use the product rule:

If f(x) and g(x) are differentiable functions and g(x) is not equal to
zero, then the derivative of the quotient ^{f}/_{g} is

which can be remembered with the saying "bottom-derivative-top minus top-derivative-bottom over bottom squared". Although a little harder to do, this result can be derived from the definition. Alternatively, the chain rule can be used to more easily establish this result.

**Example**

We can use the quotient rule to differentiate

Proceeding we have

If f(x) and g(x) are differentiable functions, then the derivative of the composition of g with f is

where the notation g'(f(x)) means the function g'(x) evaluated at f(x). Once again, this result can be established from the definition.

**Example**

Since the function h(x) = (2x^{4} + 3x)^{8} = g(f(x)) is really
the composition of the function g(x) = x^{8} with f(x) =
2x^{4} + 3x, the derivative of h is

**h'(x) = (g o f)'(x) = g'(f(x))f '(x) = 8(2x ^{4}
+ 3x)^{7}(8x^{3} + 3)**

**Example**

Differentiate

We must use all three of the rules we have established (and in the proper order no doubt). The trick is to work from the outside in. The general structure of the function a fraction; thus, we must use the quotient rule:

But, we notice that in order to carry out the differentiation of the top and bottom terms in the numerator we must use the chain and product rules, respectively. Therefore,

which upon simplification becomes

**Examples Involving Partial Derivatives**

Often when carrying out partial differentiation we must use the product, quotient, and chain rules. Several examples follow.

**Example**

Differentiate f(x,t) = cos(x t^{2}) with respect to t. We must
use the chain rule with x held constant. Thus,

**Example**

Differentiate the following function with respect to y:

To do this, we of course view x as a constant. We see that the product rule must be used since we have a product of two functions involving y.

But, in taking the derivative of each of the functions sin(x + y) and
e^(y^{2}), we must use the chain rule:

Thus,

**Example**

To differentiate

with respect to x, it appears we must use the quotient rule. This is actually not the case. Since t is held constant, the denominator is a constant; the only x term appears in the numerator. Thus, the derivative is

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**Written by Kevin
Thompson****Copyright** **©
1996 Department
of
Mathematics, Oregon State University**

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