Problem:

If 2 2 1/2 2 2 y=x(x -4a ) (x -a ) then dy/dx is?

Solution:

To find dy/dx, we need to differentiate the given function y with respect to x using the product rule and chain rule of differentiation.

Product Rule:

The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:
(d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)

In our case, the two functions are x(x-4a) and (x-a). So, we have:
y = x(x-4a)(x-a)
y' = (x-4a)(x-a)(1) + x(x-a)(1) + x(x-4a)(1)
y' = (x-4a)(x-a) + x(x-a) + x(x-4a)
y' = (x-a)(x-4a+2x) + x(x-4a)
y' = (x-a)(-2a+3x) + x(x-4a)

Chain Rule:

The chain rule states that if we have a composite function f(g(x)), then the derivative of this function is given by:
(d/dx)f(g(x)) = f'(g(x))g'(x)

In our case, we have a composite function x(x-4a)(x-a). So, we have:
f(x) = x(x-4a)(x-a)
f'(x) = 3x^2 - 10ax + 4a^2 - a

Now, we need to differentiate the outer function f(x) with respect to the inner function g(x) = x(x-4a)(x-a). So, we have:
d/dx f(g(x)) = f'(g(x))g'(x)
d/dx y = f'(x(x-4a)(x-a))(3x^2-10ax+4a^2-a)

Substituting f'(x(x-4a)(x-a)) from above, we get:
y' = (3x^2-10ax+4a^2-a)(x-a)(-2a+3x) + x(x-4a)(3x^2-10ax+4a^2-a)

Simplifying the above expression, we get:
y' = x(x-2a)(7x-10a)

Therefore, the derivative of the given function y with respect to x is y' = x(x-2a)(7x-10a).