If we can differentiate two functions f and g the derivative is

This rule can be repeated so that we can differentiate the product of more than two functions for exampe

Example

Suppose we want to differentiate

First we notice that f is the product of two functions we know how to differentiate, and sin(x).

The derivative of x² is 2x and the derivative of sin(x) is cos(x) so putting these into the rule we get

So putting in the derivative for sin and x²; we get

Proof of the Product Rule

Completely rigourus proofs of most rules with differentiation can be quite complicated and I don’t this site covers some of the concepts in enough detail, but we can give a basic outline of a proof which you can expand as you learn higher levels of maths (mainly analysis).

So as we did when deriving from first principles we want to the limit of the gradient between two point on the curve as they get closer together.

We now add and subtract f(x+t)g(x) because we can cunningly use it to get the derivatives work and because it is the same as adding 0 it is perfectly valid.

Now we bracket in such a way that we can see the derivatives we want

Spotting the limits that make dg/dx and df/dx(and making use of the fact that we can multiply limits) we get

and finally taking the limit of f(x+t) to be f(x) (since f is differentiable)

which is the required result!

Also see

• Integration by Parts (0)
• Points of Inflection (0)
• Stationary Points (Maximum and Minimums) and Differentiation (1)
• Fundamental Theorem of Calculus (0)
• Why the proof 2=1 is wrong (0)
• Differentiate Logs with Proof (0)

f(x)	f ‘(x)
c	0
	nxn-1
ln(x)	1/x
	ex
	aeax
sin(x)	cos(x)
-cos(x)	sin(x)
ln|sec(x)|	tan(x)
ln|sin(x)|	cot(x)
sinh(x)	cosh(x)
cosh(x)	sinh(x)

If there any more standard derivatives that I have missed out please leave them in the comments section

Also see

• Common Derivatives (0)
• Fundamental Theorem of Calculus (0)
• Differentiation From First Principles (with example) (1)
• Integration by Parts (0)
• Implicit Differentiation (1)
• First Order Differential Equations (6)

and

These are derived from the formula for double cos as follows

by subsitituion of

and similarly for sin²x but using the substitution for cos²x instead of sin²x.

To use these identities we simply substitute them into the integral and find the integral as normal since we know

so we get

and

Also see

• CAST Diagarams for finding values of sin, cos and tan between 0 and 360 or 2 pi (0)
• The Chain Rule (0)
• Sec, Cosec, Cot (0)
• Trigonometry Identities (11)
• Tan = sin/cos (6)
• Sine and Cos Graphs Differentiating sin and cos (7)

f(x)	f ‘(x)
c (constant)	0
xn	nxn-1
ln(x)	1/x
ex	ex
eax	aeax
sin(x)	cos(x)
cos(x)	-sin(x)
tan(x)	sec2(x)
cosec(x)	-cosec(x)cot(x)
sec(x)	sec(x)tan(x)
cot(x)	-cosec2(x)
sin-1(x)
cos-1(x)
tan-1
sinh(x)	cosh(x)
cosh(x)	sinh(x)
tanh(x)	sech2(x)
cosech(x)	-coth(x)cosech(x)
sech(x)	-tanh(x)sech(x)
coth(x)	-cosech2(x)
sinh-1(x)
cosh-1
tanh-1
a(x)b(x)	a’(x)b(x) + b’(x)a(x)
a(b(x))

If there any more standard derivatives that I have missed out please leave them in the comments section

Also see

• Common Intergrals (0)
• Fundamental Theorem of Calculus (0)
• Differentiation From First Principles (with example) (1)
• Integration by Parts (0)
• Implicit Differentiation (1)
• First Order Differential Equations (6)

These can be defined as follows,
if for all x₁ < x₂

• f(x₁) ≤ f(x₂) then f is increasing
• f(x₁ ) < f(x₁ ) then f is strictly increasing
• f(x₁ ) ≥ f(x₂) the f is decreasing
• f(x₁) < f(x₂) then f is strictly decreasing

Monotonicity and Derivatives

If a function f(x) is increasing then what we mean is that the slope is always positive, so if f is continuous we can relate the the properties of increasing and decreasing to the derivative as shown in the table below.

Its important to note that these rules only work if the function is continuous, for example consider f(x) =1/x, which is discontinuous at 0.
We can differentiate it to get f’(x) = -1/x² which we know is always negative (because the squared term is always positive) so we would expect it to be a decreasing function. However if we consider two point either side o, 1 and -1 say  we find that f is not a decreasing function because whilst -1 < 1, -1/x < 1/x contrary to our definition of decreasing

Also see

• Stationary Points (Maximum and Minimums) and Differentiation (1)
• Integrating Fractions – using the natrual logarithm – Example tan(x) (0)
• Hyperbolic Functions (0)

A point of inflection in the graph of a polynomial of degree 5

Note
While points of inflection are often found at stationary points, particularly when trying to find the nature of the stationary point, it is not a requirement that they are at one.

The number of points of infection in a graph is equal to the number of distinct real roots of the equation formed by equating the second derivative to zero.

Also see

• Stationary Points (Maximum and Minimums) and Differentiation (1)
• Product Rule for Differentiation (0)
• Fundamental Theorem of Calculus (0)
• Integration by Parts (0)
• Implicit Differentiation (1)
• First Order Differential Equations (6)

Finding the Stationary Points

We know that stationary point occur when the gradient is 0 so when the derivative of the graph is 0, so in order to find the stationary points we but first differentiate the curve.

For example lets consider the graph . We cab differentiate this to find

We must then equate the derivative to 0 and solve the resulting equation. This is because we are trying to find the points where the gradient is zero and these point occur exactly at the solutions of the equation we have formed.

So in our example we form the equation

by equating our expression for , , to 0
Solving this equation we find that stationary points occur exactly when

Note that there can be more than solution to this equation, each of which is a valid stationary point.

Finally we should also find the y co-ordinate for the stationary point by putting this value of x into the initial equation. So for this example
So the only stationary point is at

Nature of Stationary Points

The nature of a stationary point simply means what the graph is doing around it and are characterised by the second derivative, (found by differentiating the derivative). There are three types of stationary point:

1. Maximum Points: These are stationary points where the graph is sloping down on either side of the stationary point (a sad face type of curve).
   Here
2. Minimum Points: These are stationary where the graph is sloping upwards on either side of the point (a happy face)

   Here

3. Point of Inflection: Here the direction of the slope of the graph is the same either side of the stationary point, it can be in either direction.

   At a point of inflection but isn’t enough to ensure that a point really is a point of inflection as it could still be a maximum or minimum point

Checking the nature of a Stationary Point when
In this case the easiest thing to do is look a small distance either side of the point and see whether the y value is greater than or less than that of the stationary point. You can then draw yourself a picture to see what it is. For example if they are both greater than the stationary point you know it is a minimum point, but if one is greater and one is less than it is a point of inflection

Warning: checking points either side does not guarantee the correct result as there may be another stationary point or a break in the graph between where you are checking and the stationary point so you should always check using the derivatives if possible

Also see

• Points of Inflection (0)
• Product Rule for Differentiation (0)
• Monotonic (Increasing and Decreasing) Functions (0)
• Fundamental Theorem of Calculus (0)
• Integrating Fractions – using the natrual logarithm – Example tan(x) (0)
• Integration by Parts (0)

Differentiating brackets raised to a power

The chain rule can be a great short cut to differentiating brackets raised to a power as it doesn’t require you to multiply them all out, it also enables you to differentiate brackets raised to an unknown power.
Consider
This is the composite of the functions ax+b and tⁿ. So we differentiate them both to get a and nt^n-1 and then apply the formula to get

Notice how we multiplied the derivative of the inner function, a, by the derivative of the outer function nt^n-1 but substituted ax+b back in for t.

To generalise we can replace the ax+b with f(x) and by applying the above get

Differentiating Trigonometric functions

We can also use the chain rule when differentiating sin(f(x)) and cos(f(x)) since we know how to differentiate sin(x) and cos(x).
Using the chain rule we get

and

Also see

• Sine and Cos Graphs Differentiating sin and cos (7)
• CAST Diagarams for finding values of sin, cos and tan between 0 and 360 or 2 pi (0)
• Integrate cos and sin squared using double angles (2)
• Sec, Cosec, Cot (0)
• Trigonometry Identities (11)
• Tan = sin/cos (6)

if then

or in the more simple case

if then

It is this idea that allows us to know, for example,

from the knowledge that

This makes much of integration easier as it is often much easier to work out the derivative a function than work out the integral of one so we can look for functions which when differentiated give us the function that we want to integrate and then know that the integral is that function plus a constant.

Also see

• Integration by Parts (0)
• Implicit Differentiation (1)
• First Order Differential Equations (6)
• Product Rule for Differentiation (0)
• Common Intergrals (0)
• Integrate cos and sin squared using double angles (2)

and the fundamental theorem of calculus we cay say that

where c is the integration constant

Simple Example

The most basic example of this is the integration of 1/x,

More complex example: Integration of tan(x)

A slightly more complicated example of this is the integration of tan(x). To do this we must remember that and notice that . This means that -tan(x) is of the form as required. Using this we can get

Trick for using this identity

Sometimes we get integrals that are almost in this form but not exactly, eg) , however to solve these we can often factorise a constant so that it is in the required form. In this example we can take out a 2 so we get

Also see

• Integrate cos and sin squared using double angles (2)
• Monotonic (Increasing and Decreasing) Functions (0)
• Stationary Points (Maximum and Minimums) and Differentiation (1)
• Fundamental Theorem of Calculus (0)