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Differentiation from First Principles |
Algebra
Geometry
The Chain Rule
The chain rule (function of a function) is very important in differential calculus and states that:
| dy | = | dy | × | dt |
| dx | dt | dx |
(You can remember this by thinking of dy/dx as a fraction in this case (which it isn’t of course!)).
This rule allows us to differentiate a vast range of functions.
Example
If y = (1 + x²)³ , find dy/dx .
let t = 1 + x²
therefore, y = t³
dy/dt = 3t²
dt/dx = 2x
by the Chain Rule, dy/dx = dy/dt × dt/dx
so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x
= 6x(1 + x²)²
In examples such as the above one, with practise it should be possible for you
to be able to simply write down the answer without having to let t = 1 + x²
etc. This is because:
|
d |
(ax + b)n | = | a n(ax + b)n-1 |
| dx | |||
|
d |
(ax2 + b)n | = | 2ax n(ax2 + b)n-1 |
| dx | |||
|
d |
(ax3 + b)n | = | 3ax2 n(ax3 + b)n-1 |
| dx |
In other words, the differential of something in a bracket raised to the power of n is the differential of the bracket, multiplied by n times the contents of the bracket raised to the power of (n-1).
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