The Chain Rule
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.
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
Differentiate x(x² + 1)
let u = x and v = x² + 1
d (uv) = (x² + 1) + x(2x) = x² + 1 + 2x² = 3x² + 1 .
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)
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