Solve (x+7)(7/2x^frac{1{2}})

$$( x + 7 ) ( \frac { 7 } { 2 } x ^ { \frac { 1 } { 2 } } )$$

Evaluate

$$\frac{7\sqrt{x}\left(x+7\right)}{2}$$

Solution Steps

Use the distributive property to multiply x+7 by \frac{7}{2}.

$$\left(\frac{7}{2}x+\frac{49}{2}\right)x^{\frac{1}{2}}$$

Use the distributive property to multiply \frac{7}{2}x+\frac{49}{2} by x^{\frac{1}{2}}.

$$\frac{7}{2}xx^{\frac{1}{2}}+\frac{49}{2}x^{\frac{1}{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$\frac{7}{2}x^{\frac{3}{2}}+\frac{49}{2}x^{\frac{1}{2}}$$

Differentiate w.r.t. x

$$\frac{7\left(3x+7\right)}{4\sqrt{x}}$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply x+7 by \frac{7}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{7}{2}x+\frac{49}{2}\right)x^{\frac{1}{2}})$$

Use the distributive property to multiply \frac{7}{2}x+\frac{49}{2} by x^{\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7}{2}xx^{\frac{1}{2}}+\frac{49}{2}x^{\frac{1}{2}})$$

To multiply powers of the same base, add their exponents. Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7}{2}x^{\frac{3}{2}}+\frac{49}{2}x^{\frac{1}{2}})$$

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}.

$$\frac{3}{2}\times \frac{7}{2}x^{\frac{3}{2}-1}+\frac{1}{2}\times \frac{49}{2}x^{\frac{1}{2}-1}$$

Multiply \frac{3}{2} times \frac{7}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\frac{21}{4}x^{\frac{3}{2}-1}+\frac{1}{2}\times \frac{49}{2}x^{\frac{1}{2}-1}$$

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

$$\frac{21}{4}\sqrt{x}+\frac{1}{2}\times \frac{49}{2}x^{\frac{1}{2}-1}$$

Multiply \frac{1}{2} times \frac{49}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\frac{21}{4}\sqrt{x}+\frac{49}{4}x^{\frac{1}{2}-1}$$

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

$$\frac{21}{4}\sqrt{x}+\frac{49}{4}x^{-\frac{1}{2}}$$