Question

Derivatives of Products and Quotients. Find the derivative?

`y=x(x^2-2)^2`

Answer 1

`dy/dx=(x^2-2)(5x^2-2), or, 5x^4-12x^2+4`

Explanation:

`y=x(x^2-2)^2`

We first differentiate using the Product Rule.

`dy/dx=xd/dx(x^2-2)^2+(x^2-2)^2d/dx(x)`

`=xd/dx(x^2-2)^2+(x^2-2)^2`

As regards, `d/dx(x^2-2)^2`, we use the Chain Rule.

Let, `u=(x^2-2)^2=t^2, where, t=x^2-2`

So, `u=t^2, t=(x^2-2)`

Thus, `u" is a function of "t, and, t" of "x`

By the Chain Rule, then, `(du)/dx=(du)/dtdt/dx`

Now, `u=t^2 :. (du)/dt=2t, &, t=x^2-2 :. dt/dx=2x`

Utilising these, we have, `d/dx(x^2-2)^2=(du)/dx`

`=(2t)(2x)=4tx=4x(x^2-2)`

Finally, we get,

`dy/dx=4x^2(x^2-2)+(x^2-2)^2, i.e.,`

`dy/dx=(x^2-2){4x^2+(x^2-2)}=(x^2-2)(5x^2-2), or,`

`dy/dx=5x^4-12x^2+4`

Alternatively, without using the Product Rule and Chain Rule, we

can solve the Problem easily, as shown below :

`y=x(x^2-2)^2=x(x^4-4x^2+4)=x^5-4x^3+4x`

`:. dy/dx=5x^4-4(3x^2)+4=5x^4-12x^2+4,` As Before!

Enjoy Maths.!