-f(x)=x^3/(x^2+1)
To find the function f(x), we need to find the value of f(x) such that -f(x) equals x^3/(x^2+1).
First, let's substitute x^3/(x^2+1) into -f(x):
-f(x) = -(x^3/(x^2+1))
Multiplying numerator and denominator by (x^2+1), we get:
-f(x) = -[(x^3)/(x^2+1)] * [(x^2+1)/(x^2+1)]
Simplifying,
-f(x) = (-x^3(x^2+1))/(x^2+1)
Now, multiply the numerator:
-f(x) = -[(x^5+x^3)/(x^2+1)]
Finally, we can conclude that the function f(x) is:
f(x) = (x^5+x^3)/(x^2+1)
To find the derivative of the function f(x) = x^3 / (x^2 + 1), you can use the quotient rule. The quotient rule is used when you have a function that is the quotient of two other functions.
The formula for the quotient rule is:
(f/g)' = (f'g - fg') / g^2
In this case, let's call f(x) = x^3 and g(x) = (x^2 + 1).
Now, let's find the derivatives of f(x) and g(x):
f'(x) = 3x^2 (using the power rule for differentiation)
g'(x) = 2x (using the power rule for differentiation)
Now, let's use the quotient rule to find the derivative of f(x) / g(x):
(f/g)' = (f'g - fg') / g^2
= (3x^2(x^2 + 1) - x^3(2x)) / (x^2 + 1)^2
Simplifying the numerator:
= (3x^4 + 3x^2 - 2x^4) / (x^2 + 1)^2
= (x^4 + 3x^2) / (x^2 + 1)^2
Therefore, the derivative of f(x) = x^3 / (x^2 + 1) is:
f'(x) = (x^4 + 3x^2) / (x^2 + 1)^2
To find the derivative of the function f(x) = x^3/(x^2 + 1), you can use the quotient rule.
The quotient rule states that if you have a function f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, then its derivative can be calculated using the formula:
f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2
Now, let's find the derivative step by step.
Step 1: Find g(x)
In our case, g(x) = x^3.
Step 2: Find g'(x) (the derivative of g(x))
To find the derivative of x^3, apply the power rule:
g'(x) = 3x^2.
Step 3: Find h(x)
In our case, h(x) = x^2 + 1.
Step 4: Find h'(x) (the derivative of h(x))
To find the derivative of x^2 + 1, apply the power rule:
h'(x) = 2x.
Step 5: Substitute the values into the quotient rule formula
f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2
= ((x^2 + 1) * (3x^2) - (x^3) * (2x)) / (x^2 + 1)^2
Step 6: Simplify the expression
Expand the numerator and square the denominator:
f'(x) = (3x^4 + 3x^2 - 2x^4) / (x^4 + 2x^2 + 1)
= (x^4 + 3x^2) / (x^4 + 2x^2 + 1)
Therefore, the derivative of f(x) = x^3/(x^2 + 1) is f'(x) = (x^4 + 3x^2) / (x^4 + 2x^2 + 1).