Solve 4sqrt{15/8}u1/5sqrt{750}

$$4 \sqrt { \frac { 15 } { 8 } } u \frac { 1 } { 5 } \sqrt { 750 }$$

Evaluate

$$30u$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{15}{8}} as the division of square roots \frac{\sqrt{15}}{\sqrt{8}}.

$$4\times \frac{\sqrt{15}}{\sqrt{8}}u\times \frac{1}{5}\sqrt{750}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$4\times \frac{\sqrt{15}}{2\sqrt{2}}u\times \frac{1}{5}\sqrt{750}$$

Rationalize the denominator of \frac{\sqrt{15}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$4\times \frac{\sqrt{15}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}u\times \frac{1}{5}\sqrt{750}$$

The square of \sqrt{2} is 2.

$$4\times \frac{\sqrt{15}\sqrt{2}}{2\times 2}u\times \frac{1}{5}\sqrt{750}$$

To multiply \sqrt{15} and \sqrt{2}, multiply the numbers under the square root.

$$4\times \frac{\sqrt{30}}{2\times 2}u\times \frac{1}{5}\sqrt{750}$$

Multiply 2 and 2 to get 4.

$$4\times \frac{\sqrt{30}}{4}u\times \frac{1}{5}\sqrt{750}$$

Multiply 4 and \frac{1}{5} to get \frac{4}{5}.

$$\frac{4}{5}\times \frac{\sqrt{30}}{4}u\sqrt{750}$$

Factor 750=5^{2}\times 30. Rewrite the square root of the product \sqrt{5^{2}\times 30} as the product of square roots \sqrt{5^{2}}\sqrt{30}. Take the square root of 5^{2}.

$$\frac{4}{5}\times \frac{\sqrt{30}}{4}u\times 5\sqrt{30}$$

Cancel out 5 and 5.

$$4\times \frac{\sqrt{30}}{4}u\sqrt{30}$$

Cancel out 4 and 4.

$$\sqrt{30}u\sqrt{30}$$

Multiply \sqrt{30} and \sqrt{30} to get 30.

$$30u$$

Differentiate w.r.t. u

$$30$$

Steps Using Definition of a Derivative

Rewrite the square root of the division \sqrt{\frac{15}{8}} as the division of square roots \frac{\sqrt{15}}{\sqrt{8}}.

$$\frac{\mathrm{d}}{\mathrm{d}u}(4\times \frac{\sqrt{15}}{\sqrt{8}}u\times \frac{1}{5}\sqrt{750})$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}u}(4\times \frac{\sqrt{15}}{2\sqrt{2}}u\times \frac{1}{5}\sqrt{750})$$

Rationalize the denominator of \frac{\sqrt{15}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\mathrm{d}}{\mathrm{d}u}(4\times \frac{\sqrt{15}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}u\times \frac{1}{5}\sqrt{750})$$

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}u}(4\times \frac{\sqrt{15}\sqrt{2}}{2\times 2}u\times \frac{1}{5}\sqrt{750})$$

To multiply \sqrt{15} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\mathrm{d}}{\mathrm{d}u}(4\times \frac{\sqrt{30}}{2\times 2}u\times \frac{1}{5}\sqrt{750})$$

Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}u}(4\times \frac{\sqrt{30}}{4}u\times \frac{1}{5}\sqrt{750})$$

Multiply 4 and \frac{1}{5} to get \frac{4}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}u}(\frac{4}{5}\times \frac{\sqrt{30}}{4}u\sqrt{750})$$

Factor 750=5^{2}\times 30. Rewrite the square root of the product \sqrt{5^{2}\times 30} as the product of square roots \sqrt{5^{2}}\sqrt{30}. Take the square root of 5^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}u}(\frac{4}{5}\times \frac{\sqrt{30}}{4}u\times 5\sqrt{30})$$

Cancel out 5 and 5.

$$\frac{\mathrm{d}}{\mathrm{d}u}(4\times \frac{\sqrt{30}}{4}u\sqrt{30})$$

Cancel out 4 and 4.

$$\frac{\mathrm{d}}{\mathrm{d}u}(\sqrt{30}u\sqrt{30})$$

Multiply \sqrt{30} and \sqrt{30} to get 30.

$$\frac{\mathrm{d}}{\mathrm{d}u}(30u)$$

The derivative of ax^{n} is nax^{n-1}.

$$30u^{1-1}$$

Subtract 1 from 1.

$$30u^{0}$$

For any term t except 0, t^{0}=1.

$$30\times 1$$

For any term t, t\times 1=t and 1t=t.

$$30$$