Solve d/dx(2x^3/2+7x^1/2-x^-1/2)

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

Evaluate

$$\frac{\left(x+1\right)\left(6x+1\right)}{2x^{\frac{3}{2}}}$$

Steps Using Derivative Rule for Sum

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}.

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

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

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

Subtract 1 from \frac{3}{2}.

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

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

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

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

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

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

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

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

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

Differentiate w.r.t. x

$$\frac{\left(2x-3\right)\left(3x+1\right)}{4x^{\frac{5}{2}}}$$