Solve f(x)=x^frac{1{2}}(x^2+3)(x+2)

$$f ( x ) = x ^ { \frac { 1 } { 2 } } ( x ^ { 2 } + 3 ) ( x + 2 )$$

Evaluate

$$\sqrt{x}\left(x+2\right)\left(x^{2}+3\right)$$

Solution Steps

Use the distributive property to multiply x^{\frac{1}{2}} by x^{2}+3.

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

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and 2 to get \frac{5}{2}.

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

Use the distributive property to multiply x^{\frac{5}{2}}+3x^{\frac{1}{2}} by x+2.

$$x^{\frac{5}{2}}x+2x^{\frac{5}{2}}+3x^{\frac{1}{2}}x+6x^{\frac{1}{2}}$$

To multiply powers of the same base, add their exponents. Add \frac{5}{2} and 1 to get \frac{7}{2}.

$$x^{\frac{7}{2}}+2x^{\frac{5}{2}}+3x^{\frac{1}{2}}x+6x^{\frac{1}{2}}$$

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

$$x^{\frac{7}{2}}+2x^{\frac{5}{2}}+3x^{\frac{3}{2}}+6x^{\frac{1}{2}}$$

Differentiate w.r.t. x

$$\frac{\left(x+1\right)\left(7x^{2}+3x+6\right)}{2\sqrt{x}}$$