Solve -3x^-1/2+2x^1/2+5x^1/2

$$- 3 x ^ { - 1 / 2 } + 2 x ^ { 1 / 2 } + 5 x ^ { 1 / 2 }$$

Differentiate w.r.t. x

$$\frac{7x+3}{2x^{\frac{3}{2}}}$$

Steps Using Derivative Rule for Sum

Combine 2x^{\frac{1}{2}} and 5x^{\frac{1}{2}} to get 7x^{\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-3x^{-\frac{1}{2}}+7x^{\frac{1}{2}})$$

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}.

$$-\frac{1}{2}\left(-3\right)x^{-\frac{1}{2}-1}+\frac{1}{2}\times 7x^{\frac{1}{2}-1}$$

Multiply -\frac{1}{2} times -3.

$$\frac{3}{2}x^{-\frac{1}{2}-1}+\frac{1}{2}\times 7x^{\frac{1}{2}-1}$$

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

$$\frac{3}{2}x^{-\frac{3}{2}}+\frac{1}{2}\times 7x^{\frac{1}{2}-1}$$

Multiply \frac{1}{2} times 7.

$$\frac{3}{2}x^{-\frac{3}{2}}+\frac{7}{2}x^{\frac{1}{2}-1}$$

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

$$\frac{3}{2}x^{-\frac{3}{2}}+\frac{7}{2}x^{-\frac{1}{2}}$$

Evaluate

$$\frac{7x-3}{\sqrt{x}}$$

Solution Steps

Combine 2x^{\frac{1}{2}} and 5x^{\frac{1}{2}} to get 7x^{\frac{1}{2}}.

$$-3x^{-\frac{1}{2}}+7x^{\frac{1}{2}}$$