Question

find dy/dx if y=a^2x + x/2x+1​

Answer 1

Here chain rule will be applied…

Consider (2x+1) = a

Now differentiate on both sides with respect to ‘a’, we get,

2dx/da = 1

So, dx/da = 1/2 …(Eqn. 1)

Now, put (2x+1)=a in the main equation, you’ll get

y= a^5

Differentiating on both sides with respect to ‘a’ we get,

dy/da= 5a^4 …(Eqn. 2)

Now divide (Eqn. 2) by (Eqn. 1), we get

(dy/da)/(dx/da) =5a^4/(1/2)

So, dy/dx= 10a^4

Resubstituting value of ‘a’, we get

dy/dx= 10(2x+1)^4

Method 2:

This is also the chain rule but with less steps:

Consider (2x+1) is any variable and differentiate it the same way as you would differentiate y=x^n

Which is dy/dx = n*x^(n-1)

So differentiation the main equation we get,

dy/dx= 5*(2x+1)^4 * d(2x+1)/dx

[The point to be noted here is after differentiating the bracket considering it as a variable, we need to again mulply with the differential of the bracket wrt x]

So we get,

dy/dx= 10*(2x+1)^4

And that’s your answer…

Do remember the following Chain Rule:

dy/dx= (dy/da)* (da/dx)

Hope it helps…

Feel good :)