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

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

Differentiate w.r.t. u

$$-2\sqrt{v}u-15\sqrt{\frac{v}{u}}+6$$

Steps Using Derivative Rule for Sum

Multiply -1 and 30 to get -30.

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

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}.

$$6u^{1-1}+\frac{1}{2}\left(-30\sqrt{v}\right)u^{\frac{1}{2}-1}+2\left(-\sqrt{v}\right)u^{2-1}$$

Subtract 1 from 1.

$$6u^{0}+\frac{1}{2}\left(-30\sqrt{v}\right)u^{\frac{1}{2}-1}+2\left(-\sqrt{v}\right)u^{2-1}$$

Multiply \frac{1}{2} times -30v^{\frac{1}{2}}.

$$6u^{0}+\left(-15\sqrt{v}\right)u^{\frac{1}{2}-1}+2\left(-\sqrt{v}\right)u^{2-1}$$

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

$$6u^{0}+\left(-15\sqrt{v}\right)u^{-\frac{1}{2}}+2\left(-\sqrt{v}\right)u^{2-1}$$

Multiply \frac{1}{2} times -30v^{\frac{1}{2}}.

$$6u^{0}+\left(-15\sqrt{v}\right)u^{-\frac{1}{2}}+\left(-2\sqrt{v}\right)u^{2-1}$$

Subtract 1 from 2.

$$6u^{0}+\left(-15\sqrt{v}\right)u^{-\frac{1}{2}}+\left(-2\sqrt{v}\right)u^{1}$$

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

$$6u^{0}+\left(-15\sqrt{v}\right)u^{-\frac{1}{2}}+\left(-2\sqrt{v}\right)u$$

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

$$6\times 1+\left(-15\sqrt{v}\right)u^{-\frac{1}{2}}+\left(-2\sqrt{v}\right)u$$

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

$$6+\left(-15\sqrt{v}\right)u^{-\frac{1}{2}}+\left(-2\sqrt{v}\right)u$$

Evaluate

$$-\sqrt{v}u^{2}-30\sqrt{uv}+6u+5$$