Chain, Product and Quotient
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The chain rule is very important in differential calculus and states that:
dy  =  dy  ×  dt
dx       dt      dx

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:

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-1) multiplied by the contents of the bracket raised to the power of (n-1).

The Product Rule
This is another very useful formula:

d (uv) = vdu + udv
dx            dx      dx

Example:
Differentiate x(x² + 1)
let u = x and v = x² + 1
d (uv) = (x² + 1) + x(2x) = x² + 1 + 2x² = 3x² + 1 .
dx

Again, with practise you shouldn't have to write out u = ... and v = ... every time.

The Quotient Rule

d (u/v)  = v(du/dx) - u(dv/dx)
dx                         v²

Example:
If y =    x³    , find dy/dx
         x + 4
Let u = x³ and v = (x + 4). Using the quotient rule, dy/dx =
(x + 4)(3x²) - x³(1)  =   2x³ + 12x²
      (x + 4)²                     (x + 4)²

© Matthew Pinkey

Other Notes in this Category

1. Chain, Product and Quotient
2. Differential Equations
3. Differentiation
4. Differentiation from 1st Principles
5. Differentiation of Trig Funcns
6. Exponentials & Logarithms
7. Implicit Differentiation
8. Integration
9. Integration by Parts
10. Integration by Substitution
11. Tangents and Normals
12. The Second Derivative
13. Uses of Differentiation
14. Volumes of Revolution

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