Solve f(x)=x^3/2(x^2+3)(x+2)

$$f ( x ) = x ^ { 3 / 2 } ( x ^ { 2 } + 3 ) ( x + 2 )$$

Evaluate

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

Solution Steps

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

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

To multiply powers of the same base, add their exponents. Add \frac{3}{2} and 2 to get \frac{7}{2}.

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

Use the distributive property to multiply x^{\frac{7}{2}}+3x^{\frac{3}{2}} by x+2.

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

To multiply powers of the same base, add their exponents. Add \frac{7}{2} and 1 to get \frac{9}{2}.

$$x^{\frac{9}{2}}+2x^{\frac{7}{2}}+3x^{\frac{3}{2}}x+6x^{\frac{3}{2}}$$

To multiply powers of the same base, add their exponents. Add \frac{3}{2} and 1 to get \frac{5}{2}.

$$x^{\frac{9}{2}}+2x^{\frac{7}{2}}+3x^{\frac{5}{2}}+6x^{\frac{3}{2}}$$

Differentiate w.r.t. x

$$\frac{\sqrt{x}\left(9x^{3}+14x^{2}+15x+18\right)}{2}$$