Solve 6u-30{u}^frac{1{2}}{v}^frac{1{2}}-{u}^frac{1{2}}{v}^frac{1{2}}+5

$$6u-30 { u }^{ \frac{ 1 }{ 2 } } { v }^{ \frac{ 1 }{ 2 } } - { u }^{ \frac{ 1 }{ 2 } } { v }^{ \frac{ 1 }{ 2 } } +5$$

Differentiate w.r.t. u

$$-\frac{31\sqrt{\frac{v}{u}}}{2}+6$$

Steps Using Derivative Rule for Sum

Multiply -1 and 30 to get -30.

$$\frac{\mathrm{d}}{\mathrm{d}u}(6u-30u^{\frac{1}{2}}v^{\frac{1}{2}}-u^{\frac{1}{2}}v^{\frac{1}{2}}+5)$$

Combine -30u^{\frac{1}{2}}v^{\frac{1}{2}} and -u^{\frac{1}{2}}v^{\frac{1}{2}} to get -31u^{\frac{1}{2}}v^{\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}u}(6u-31u^{\frac{1}{2}}v^{\frac{1}{2}}+5)$$

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}.

$$6u^{1-1}+\frac{1}{2}\left(-31\sqrt{v}\right)u^{\frac{1}{2}-1}$$

Subtract 1 from 1.

$$6u^{0}+\frac{1}{2}\left(-31\sqrt{v}\right)u^{\frac{1}{2}-1}$$

Multiply \frac{1}{2} times -31v^{\frac{1}{2}}.

$$6u^{0}+\left(-\frac{31\sqrt{v}}{2}\right)u^{\frac{1}{2}-1}$$

Subtract 1 from \frac{1}{2}.

$$6u^{0}+\left(-\frac{31\sqrt{v}}{2}\right)u^{-\frac{1}{2}}$$

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

$$6\times 1+\left(-\frac{31\sqrt{v}}{2}\right)u^{-\frac{1}{2}}$$

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

$$6+\left(-\frac{31\sqrt{v}}{2}\right)u^{-\frac{1}{2}}$$

Evaluate

$$-31\sqrt{uv}+6u+5$$