Find the 4th Derivative e^(x^2)

To apply the Chain Rule, set ${\displaystyle u}$ as ${\displaystyle x^2}$.

Differentiate using the Exponential Rule which states that ${\displaystyle \frac{d}{du}\left[a^u\right]}$ is ${\displaystyle a^u\ln \left(a\right)}$ where ${\displaystyle a}$=${\displaystyle e}$.

Replace all occurrences of ${\displaystyle u}$ with ${\displaystyle x^2}$.

Reorder the factors of ${\displaystyle e^{x^2}\left(2x\right)}$.

Reorder factors in ${\displaystyle 2e^{x^2}x}$.

To apply the Chain Rule, set ${\displaystyle u}$ as ${\displaystyle x^2}$.

Differentiate using the Exponential Rule which states that ${\displaystyle \frac{d}{du}\left[a^u\right]}$ is ${\displaystyle a^u\ln \left(a\right)}$ where ${\displaystyle a}$=${\displaystyle e}$.

Replace all occurrences of ${\displaystyle u}$ with ${\displaystyle x^2}$.

Add ${\displaystyle 1}$ and ${\displaystyle 1}$.

Move ${\displaystyle 2}$ to the left of ${\displaystyle e^{x^2}}$.

Apply the distributive property.

Multiply ${\displaystyle 2}$ by ${\displaystyle 2}$.

Reorder terms.

Reorder factors in ${\displaystyle 4e^{x^2}{x}^2+2e^{x^2}}$.

Since ${\displaystyle 4}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle 4x^2{e}^{x^2}}$ with respect to ${\displaystyle x}$ is ${\displaystyle 4\frac{d}{dx}\left[x^2{e}^{x^2}\right]}$.

Differentiate using the Product Rule which states that ${\displaystyle \frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]}$ is ${\displaystyle f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]}$ where ${\displaystyle f\left(x\right)=x^2}$ and ${\displaystyle g\left(x\right)=e^{x^2}}$.

Differentiate using the chain rule, which states that ${\displaystyle \frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]}$ is ${\displaystyle f\prime \left(g\left(x\right)\right)g\prime \left(x\right)}$ where ${\displaystyle f\left(x\right)=e^x}$ and ${\displaystyle g\left(x\right)=x^2}$.

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dx}\left[x^n\right]}$ is ${\displaystyle n{x}^{n-1}}$ where ${\displaystyle n=2}$.

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dx}\left[x^n\right]}$ is ${\displaystyle n{x}^{n-1}}$ where ${\displaystyle n=2}$.

Multiply ${\displaystyle x^2}$ by ${\displaystyle x}$ by adding the exponents.

Move ${\displaystyle 2}$ to the left of ${\displaystyle e^{x^2}}$.

Since ${\displaystyle 2}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle 2e^{x^2}}$ with respect to ${\displaystyle x}$ is ${\displaystyle 2\frac{d}{dx}\left[e^{x^2}\right]}$.

Differentiate using the chain rule, which states that ${\displaystyle \frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]}$ is ${\displaystyle f\prime \left(g\left(x\right)\right)g\prime \left(x\right)}$ where ${\displaystyle f\left(x\right)=e^x}$ and ${\displaystyle g\left(x\right)=x^2}$.

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dx}\left[x^n\right]}$ is ${\displaystyle n{x}^{n-1}}$ where ${\displaystyle n=2}$.

Multiply ${\displaystyle 2}$ by ${\displaystyle 2}$.

Apply the distributive property.

Combine terms.

Reorder terms.

Reorder factors in ${\displaystyle 8e^{x^2}{x}^3+12e^{x^2}x}$.

Since ${\displaystyle 8}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle 8x^3{e}^{x^2}}$ with respect to ${\displaystyle x}$ is ${\displaystyle 8\frac{d}{dx}\left[x^3{e}^{x^2}\right]}$.

Differentiate using the Product Rule which states that ${\displaystyle \frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]}$ is ${\displaystyle f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]}$ where ${\displaystyle f\left(x\right)=x^3}$ and ${\displaystyle g\left(x\right)=e^{x^2}}$.

Differentiate using the chain rule, which states that ${\displaystyle \frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]}$ is ${\displaystyle f\prime \left(g\left(x\right)\right)g\prime \left(x\right)}$ where ${\displaystyle f\left(x\right)=e^x}$ and ${\displaystyle g\left(x\right)=x^2}$.

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dx}\left[x^n\right]}$ is ${\displaystyle n{x}^{n-1}}$ where ${\displaystyle n=2}$.

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dx}\left[x^n\right]}$ is ${\displaystyle n{x}^{n-1}}$ where ${\displaystyle n=3}$.

Multiply ${\displaystyle x^3}$ by ${\displaystyle x}$ by adding the exponents.

Move ${\displaystyle 2}$ to the left of ${\displaystyle e^{x^2}}$.

Since ${\displaystyle 12}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle 12x{e}^{x^2}}$ with respect to ${\displaystyle x}$ is ${\displaystyle 12\frac{d}{dx}\left[x{e}^{x^2}\right]}$.

Differentiate using the Product Rule which states that ${\displaystyle \frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]}$ is ${\displaystyle f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]}$ where ${\displaystyle f\left(x\right)=x}$ and ${\displaystyle g\left(x\right)=e^{x^2}}$.

Differentiate using the chain rule, which states that ${\displaystyle \frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]}$ is ${\displaystyle f\prime \left(g\left(x\right)\right)g\prime \left(x\right)}$ where ${\displaystyle f\left(x\right)=e^x}$ and ${\displaystyle g\left(x\right)=x^2}$.