Introduction Of Chain Rule Of Partial Derivative

 An introduction of chain rule of partial derivative, let's first talk about one, two and three variables.

Example; CPD Of One Variable f(x)

f ⇔ x ⇔ t = df/dx = df/dx * dx/dt = formulae

Eg

f(x) = x²     where  x = t + 1

 

Solution 

df/dx = 2x 

dx/dt = (t + 1)^2 = 2(t + 1)*1

= 2t + 2

Now by using the above formulae we can arrange it as.

2x * 2t + 2 

But  x = t +1, therefore 2(t + 1) * 2t + 2

2t + 2 * 2t + 2

4t + 4  hence proved.

Example ; CPD Of Two Variable f(x, y)

Is illustrated by  f ⇔ x ⇔ t ⇔y

 

f(x)= df/dt = df/dx * dx/dt + df/dy * dy/dt 

eg ; f(x, y) = x + y² . where x = t² + 1 and y = ¹/ₜ

 Find df/dt using CPD

Solution

 f(x) = x and f'(x) = 1 , f(y) = y² and f'(y) = 2y

 df/dx = 1, dx/dt = 2t

df/dy = 2y, dy/dt = -t⁻²

By using the above mentioned formulae we have ;

2t * 1 + 2y * -t⁻²

2t + 2(1/t) * - t⁻²

2t  - 2 (1/t³)

2t - 2/t³

 

Note ; this solution was generated using one of the law of Indices.

That's how it keeps going even if you have three variables like f(x, y, z). Use desame method and guidance, if there is any enquiry please let us know using contact us page.