Solve 1/n+1/frac{1{2}n^-frac{1{2}}}

$$\frac { \frac { 1 } { n + 1 } } { \frac { 1 } { 2 } n ^ { - \frac { 1 } { 2 } } }$$

Evaluate

$$\frac{2\sqrt{n}}{n+1}$$

Solution Steps

Express \frac{\frac{1}{n+1}}{\frac{1}{2}n^{-\frac{1}{2}}} as a single fraction.

$$\frac{1}{\left(n+1\right)\times \frac{1}{2}n^{-\frac{1}{2}}}$$

Use the distributive property to multiply n+1 by \frac{1}{2}.

$$\frac{1}{\left(\frac{1}{2}n+\frac{1}{2}\right)n^{-\frac{1}{2}}}$$

Use the distributive property to multiply \frac{1}{2}n+\frac{1}{2} by n^{-\frac{1}{2}}.

$$\frac{1}{\frac{1}{2}nn^{-\frac{1}{2}}+\frac{1}{2}n^{-\frac{1}{2}}}$$

To multiply powers of the same base, add their exponents. Add 1 and -\frac{1}{2} to get \frac{1}{2}.

$$\frac{1}{\frac{1}{2}n^{\frac{1}{2}}+\frac{1}{2}n^{-\frac{1}{2}}}$$

Differentiate w.r.t. n

$$\frac{1-n}{\sqrt{n}\left(n+1\right)^{2}}$$

Steps Using Chain Rule

Express \frac{\frac{1}{n+1}}{\frac{1}{2}n^{-\frac{1}{2}}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{\left(n+1\right)\times \frac{1}{2}n^{-\frac{1}{2}}})$$

Use the distributive property to multiply n+1 by \frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{\left(\frac{1}{2}n+\frac{1}{2}\right)n^{-\frac{1}{2}}})$$

Use the distributive property to multiply \frac{1}{2}n+\frac{1}{2} by n^{-\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{\frac{1}{2}nn^{-\frac{1}{2}}+\frac{1}{2}n^{-\frac{1}{2}}})$$

To multiply powers of the same base, add their exponents. Add 1 and -\frac{1}{2} to get \frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{\frac{1}{2}n^{\frac{1}{2}}+\frac{1}{2}n^{-\frac{1}{2}}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(\frac{1}{2}\sqrt{n}+\frac{1}{2}n^{-\frac{1}{2}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{2}\sqrt{n}+\frac{1}{2}n^{-\frac{1}{2}})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(\frac{1}{2}\sqrt{n}+\frac{1}{2}n^{-\frac{1}{2}}\right)^{-2}\left(\frac{1}{2}\times \frac{1}{2}n^{\frac{1}{2}-1}-\frac{1}{2}\times \frac{1}{2}n^{-\frac{1}{2}-1}\right)$$

Simplify.

$$\left(\frac{1}{2}\sqrt{n}+\frac{1}{2}n^{-\frac{1}{2}}\right)^{-2}\left(-\frac{1}{4}n^{-\frac{1}{2}}+\frac{1}{4}n^{-\frac{3}{2}}\right)$$