find the derivative
f(x)= (x^4+x^2+1)/(x^2+1)
To find the derivative of the given function f(x) = (x^4 + x^2 + 1)/(x^2 + 1), we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = u(x)/v(x), where u(x) and v(x) are differentiable functions, then the derivative of f(x) is given by:
f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2
Let's break down the steps to find the derivative:
Step 1: Determine u(x) and v(x)
In our case, u(x) = (x^4 + x^2 + 1) and v(x) = (x^2 + 1).
Step 2: Find u'(x) and v'(x)
To find u'(x), we differentiate u(x) term by term:
u'(x) = d/dx (x^4 + x^2 + 1)
= 4x^3 + 2x + 0
= 4x^3 + 2x
To find v'(x), we differentiate v(x) term by term:
v'(x) = d/dx (x^2 + 1)
= 2x + 0
= 2x
Step 3: Apply the quotient rule
Using the quotient rule formula, substitute the values we found into the formula:
f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2
= [(4x^3 + 2x)(x^2 + 1) - (x^4 + x^2 + 1)(2x)] / [(x^2 + 1)^2]
= [4x^5 + 6x^3 + 2x - 2x^5 - 2x - 2] / [(x^2 + 1)^2]
= [2x^5 + 6x^3 - 2] / [(x^2 + 1)^2]
Therefore, the derivative of f(x) = (x^4 + x^2 + 1)/(x^2 + 1) is f'(x) = [2x^5 + 6x^3 - 2] / [(x^2 + 1)^2].