# Find Derivative of f(x) = x/(x+c/x): Step-by-Step

• nothing123
In summary, the derivative of f(x) = x/(x+c/x) is incorrect. The correct method would be to either use the quotient rule or the product rule after simplifying the original expression.
nothing123
find the derivative of the following:
f(x) = x/(x+c/x)

through simplication, i got:
x(x+cx^-1)^-1
=x(x^-1 + c^-1*x)
=1+1/c*x^2

taking the derivative,
-c^-2*x^2 + c^-1*2x

its wrong though so where did i go wrong?

nothing123 said:
through simplication, i got:
x(x+cx^-1)^-1
=x(x^-1 + c^-1*x)
=1+1/c*x^2

That's not right. (x+y)-1 is not the same as x-1 + y-1.

Without simplifying the original expression you could have used the quotient rule to find the derivative, OR you could differentiate x(x+cx^-1)^-1 using the product rule.

I would be inclined to multiply both numerator and denominator of the function by x:
$$f(x)= \frac{x}{x+ \frac{c}{x}}= \frac{x^2}{x^2+ c}$$
for all x except 0.

## 1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a particular point. It is the slope of the tangent line to the function at that point.

## 2. Why is it important to find the derivative of a function?

Finding the derivative of a function is important because it helps us to understand the behavior of the function and its rate of change. It also allows us to find maximum and minimum points, which are crucial in many real-world applications.

## 3. What is the quotient rule for finding derivatives?

The quotient rule is a formula used to find the derivative of a function that is expressed as the quotient of two other functions. It states that the derivative of f(x)/g(x) is equal to (f'(x)g(x) - g'(x)f(x)) / (g(x))^2.

## 4. How do I apply the quotient rule to find the derivative of f(x) = x/(x+c/x)?

To apply the quotient rule to this function, we first need to rewrite it as f(x) = x(x+c/x)^-1. Then, using the quotient rule, we get f'(x) = [(x+c/x)^-1 - (-x(x+c/x)^-2)(1+c/x^2)] / (x+c/x)^2. We can simplify this expression further to get f'(x) = (1+c/x^2) / (x+c/x)^2.

## 5. What is the step-by-step process for finding the derivative of f(x) = x/(x+c/x)?

The step-by-step process for finding the derivative of this function would be:
1. Rewrite the function as f(x) = x(x+c/x)^-1.
2. Apply the quotient rule, which gives f'(x) = [(x+c/x)^-1 - (-x(x+c/x)^-2)(1+c/x^2)] / (x+c/x)^2.
3. Simplify the expression to get f'(x) = (1+c/x^2) / (x+c/x)^2.
4. Further simplify the expression if possible.
5. The final result is the derivative of f(x) = x/(x+c/x).

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