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