DIFERENTIATE THE FUNCTION:
y=[(x^2)+1][(1-x^2)]
You can simplify that by recognizing that
[(x^2)+1][(1-x^2)] = 1 - x^4
That should be easy for you to differentiate.
The derivative is just -4x^3
thanks!
To differentiate the function y = [(x^2) + 1][(1 - x^2)], we can use the product rule of differentiation, which 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 this case, u(x) = (x^2) + 1 and v(x) = (1 - x^2). Let's find the derivatives of both u(x) and v(x) first.
To differentiate u(x), we apply the power rule, which states that the derivative of x^n with respect to x is nx^(n-1):
u'(x) = d/dx[(x^2) + 1]
= 2x
To differentiate v(x), we can rewrite it as (1 - x^2) = 1 - x^2 = -x^2 + 1. Then we apply the power rule as before:
v'(x) = d/dx[-x^2 + 1]
= -2x
Now, substituting u'(x) and v'(x) back into the product rule formula, we have:
(d/dx)[(x^2) + 1][(1 - x^2)] = (2x)(1 - x^2) + [(x^2) + 1](-2x)
Expanding and simplifying the expression:
= 2x - 2x^3 + (-2x^3) - 2x
= 4x - 4x^3
Therefore, the derivative of the given function y = [(x^2) + 1][(1 - x^2)] is dy/dx = 4x - 4x^3.