# How do you differentiate #f(x)=x/(1+sqrtx)#?

By signing up, you agree to our Terms of Service and Privacy Policy

To differentiate ( f(x) = \frac{x}{1 + \sqrt{x}} ), you can use the quotient rule, which states that if ( u(x) ) and ( v(x) ) are differentiable functions, then the derivative of ( \frac{u(x)}{v(x)} ) is given by:

[ \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ]

For ( f(x) = \frac{x}{1 + \sqrt{x}} ), let ( u(x) = x ) and ( v(x) = 1 + \sqrt{x} ). Then, apply the quotient rule:

[ f'(x) = \frac{(1 + \sqrt{x}) - \frac{1}{2\sqrt{x}}(x)}{(1 + \sqrt{x})^2} ]

[ f'(x) = \frac{1 + \sqrt{x} - \frac{x}{2\sqrt{x}}}{(1 + \sqrt{x})^2} ]

[ f'(x) = \frac{1 + \sqrt{x} - \frac{1}{2}\sqrt{x}}{(1 + \sqrt{x})^2} ]

[ f'(x) = \frac{1 + \frac{1}{2}\sqrt{x}}{(1 + \sqrt{x})^2} ]

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7