Basic differentiation

Finding the gradient of a straight line is easy: .  Finding the gradient of a curve isn’t as easy.  Drawing a tangent and finding its gradient is one solution, but there’s a better way than that- differentiation.

Differentiating, , a function, , results in a derivative function, :

To differentiate an entire function, you must differentiate each term individually- and to do this, you follow a simple rule:

Example

You can’t differentiate and directly, so it’s easier to deal with fractions and other algebraic expressions first and convert them into form:

Now we can differentiate:

And simplify:

And that’s it.

Explanation

But just in case, here’s a brief run-down for each expression at a time…

• because .

This is just a straight-forward differentiation here- nothing really to say about it.

• because is just .

Differentiating a quadratic expression leaves x to the power of 1, so there’s no point in writing the index down.

• because ;
• for the same reason.

When you differentiate x to the power of 1, you create x to the power of 0.  Anything to the power of 0 always equals 1:

• .

The square root of x can be re-written as x to the power of a half.  Minus 1 from 0.5 and you get -0.5.

• because .

Re-write fractions as negative indices, and then it’s exactly the same as the first expression, only it’s negative this time (both constant and index).

• because .

When you have only a constant on its own, actually x is to the power of 0- it’s just not shown.  Since you multiply the index by the constant, you’re left with 0: