Solve 1/sqrt{2pi}exp{-x^2/2}*1/sqrt{2pi}exp{-y^2/2}

$$\frac { 1 } { \sqrt { 2 \pi } } \exp \{ - \frac { x ^ { 2 } } { 2 } \} \cdot \frac { 1 } { \sqrt { 2 \pi } } \exp \{ - \frac { y ^ { 2 } } { 2 } \}$$

Evaluate

$$\frac{x^{4}\left(epy\right)^{2}}{8\pi }$$

Short Solution Steps

Multiply \frac{1}{\sqrt{2\pi }} and \frac{1}{\sqrt{2\pi }} to get \left(\frac{1}{\sqrt{2\pi }}\right)^{2}.

$$\left(\frac{1}{\sqrt{2\pi }}\right)^{2}exp\left(-\frac{x^{2}}{2}\right)exp\left(-\frac{y^{2}}{2}\right)$$

Multiply e and e to get e^{2}.

$$\left(\frac{1}{\sqrt{2\pi }}\right)^{2}e^{2}xp\left(-\frac{x^{2}}{2}\right)xp\left(-\frac{y^{2}}{2}\right)$$

Multiply x and x to get x^{2}.

$$\left(\frac{1}{\sqrt{2\pi }}\right)^{2}e^{2}x^{2}p\left(-\frac{x^{2}}{2}\right)p\left(-\frac{y^{2}}{2}\right)$$

Multiply p and p to get p^{2}.

$$\left(\frac{1}{\sqrt{2\pi }}\right)^{2}e^{2}x^{2}p^{2}\left(-\frac{x^{2}}{2}\right)\left(-\frac{y^{2}}{2}\right)$$

To raise \frac{1}{\sqrt{2\pi }} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{2}}{\left(\sqrt{2\pi }\right)^{2}}e^{2}x^{2}p^{2}\left(-\frac{x^{2}}{2}\right)\left(-\frac{y^{2}}{2}\right)$$

Express \frac{1^{2}}{\left(\sqrt{2\pi }\right)^{2}}e^{2} as a single fraction.

$$\frac{1^{2}e^{2}}{\left(\sqrt{2\pi }\right)^{2}}x^{2}p^{2}\left(-\frac{x^{2}}{2}\right)\left(-\frac{y^{2}}{2}\right)$$

Express \frac{1^{2}e^{2}}{\left(\sqrt{2\pi }\right)^{2}}x^{2} as a single fraction.

$$\frac{1^{2}e^{2}x^{2}}{\left(\sqrt{2\pi }\right)^{2}}p^{2}\left(-\frac{x^{2}}{2}\right)\left(-\frac{y^{2}}{2}\right)$$

Express \frac{1^{2}e^{2}x^{2}}{\left(\sqrt{2\pi }\right)^{2}}p^{2} as a single fraction.

$$\frac{1^{2}e^{2}x^{2}p^{2}}{\left(\sqrt{2\pi }\right)^{2}}\left(-\frac{x^{2}}{2}\right)\left(-\frac{y^{2}}{2}\right)$$

Multiply \frac{1^{2}e^{2}x^{2}p^{2}}{\left(\sqrt{2\pi }\right)^{2}} times -\frac{x^{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-1^{2}e^{2}x^{2}p^{2}x^{2}}{\left(\sqrt{2\pi }\right)^{2}\times 2}\left(-\frac{y^{2}}{2}\right)$$

Multiply \frac{-1^{2}e^{2}x^{2}p^{2}x^{2}}{\left(\sqrt{2\pi }\right)^{2}\times 2} times -\frac{y^{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\left(-1\right)\times 1^{2}e^{2}x^{2}p^{2}x^{2}y^{2}}{\left(\sqrt{2\pi }\right)^{2}\times 2\times 2}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{-\left(-1\right)\times 1^{2}e^{2}x^{4}p^{2}y^{2}}{\left(\sqrt{2\pi }\right)^{2}\times 2\times 2}$$

Multiply -1 and -1 to get 1.

$$\frac{1^{2}e^{2}x^{4}p^{2}y^{2}}{\left(\sqrt{2\pi }\right)^{2}\times 2\times 2}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{1e^{2}x^{4}p^{2}y^{2}}{\left(\sqrt{2\pi }\right)^{2}\times 2\times 2}$$

Calculate \sqrt{2\pi } to the power of 2 and get 2\pi .

$$\frac{1e^{2}x^{4}p^{2}y^{2}}{2\pi \times 2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{1e^{2}x^{4}p^{2}y^{2}}{4\pi \times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{1e^{2}x^{4}p^{2}y^{2}}{8\pi }$$

Cancel out 1 in both numerator and denominator.

$$\frac{e^{2}x^{4}p^{2}y^{2}}{8\pi }$$

Differentiate w.r.t. x

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