Solve (2)Gamma(3)Gamma(3/2)/Gamma(frac{9{2})}

$$( 2 ) \frac { \Gamma ( 3 ) \Gamma ( \frac { 3 } { 2 } ) } { \Gamma ( \frac { 9 } { 2 } ) }$$

Evaluate

$$2\Gamma$$

Solution Steps

Multiply \Gamma and \Gamma to get \Gamma ^{2}.

$$2\times \frac{\Gamma ^{2}\times 3\times \frac{3}{2}}{\Gamma \times \frac{9}{2}}$$

Cancel out \Gamma in both numerator and denominator.

$$2\times \frac{\frac{3}{2}\times 3\Gamma }{\frac{9}{2}}$$

Express \frac{3}{2}\times 3 as a single fraction.

$$2\times \frac{\frac{3\times 3}{2}\Gamma }{\frac{9}{2}}$$

Multiply 3 and 3 to get 9.

$$2\times \frac{\frac{9}{2}\Gamma }{\frac{9}{2}}$$

Divide \frac{9}{2}\Gamma by \frac{9}{2} by multiplying \frac{9}{2}\Gamma by the reciprocal of \frac{9}{2}.

$$2\times \frac{\frac{9}{2}\Gamma \times 2}{9}$$

Cancel out 2 and 2.

$$2\times \frac{9\Gamma }{9}$$

Cancel out 9 and 9.

$$2\Gamma$$

Differentiate w.r.t. Γ

$$2$$

Steps Using Definition of a Derivative

Multiply \Gamma and \Gamma to get \Gamma ^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}\Gamma }(2\times \frac{\Gamma ^{2}\times 3\times \frac{3}{2}}{\Gamma \times \frac{9}{2}})$$

Cancel out \Gamma in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}\Gamma }(2\times \frac{\frac{3}{2}\times 3\Gamma }{\frac{9}{2}})$$

Express \frac{3}{2}\times 3 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}\Gamma }(2\times \frac{\frac{3\times 3}{2}\Gamma }{\frac{9}{2}})$$

Multiply 3 and 3 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}\Gamma }(2\times \frac{\frac{9}{2}\Gamma }{\frac{9}{2}})$$

Divide \frac{9}{2}\Gamma by \frac{9}{2} by multiplying \frac{9}{2}\Gamma by the reciprocal of \frac{9}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}\Gamma }(2\times \frac{\frac{9}{2}\Gamma \times 2}{9})$$

Cancel out 2 and 2.

$$\frac{\mathrm{d}}{\mathrm{d}\Gamma }(2\times \frac{9\Gamma }{9})$$

Cancel out 9 and 9.

$$\frac{\mathrm{d}}{\mathrm{d}\Gamma }(2\Gamma )$$

The derivative of ax^{n} is nax^{n-1}.

$$2\Gamma ^{1-1}$$

Subtract 1 from 1.

$$2\Gamma ^{0}$$

For any term t except 0, t^{0}=1.

$$2\times 1$$

For any term t, t\times 1=t and 1t=t.

$$2$$