Find dy/dx arctan(5x^2y)=x+4xy^2

To apply the Chain Rule, set ${\displaystyle u}$ as ${\displaystyle 5x^{2}y}$.

The derivative of ${\displaystyle \arctan \left(u\right)}$ with respect to ${\displaystyle u}$ is ${\displaystyle \frac{1}{1+u^2}}$.

Replace all occurrences of ${\displaystyle u}$ with ${\displaystyle 5x^{2}y}$.

Factor ${\displaystyle 5}$ out of ${\displaystyle 5x^{2}y}$.

Simplify the expression.

Since ${\displaystyle 5}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle 5x^{2}y}$ with respect to ${\displaystyle x}$ is ${\displaystyle 5\frac{d}{dx}\left[x^{2}y\right]}$.

Combine ${\displaystyle 5}$ and ${\displaystyle \frac{1}{1+25{\left(x^{2}y\right)}^2}}$.

Apply the product rule to ${\displaystyle x^{2}y}$.

Multiply the exponents in ${\displaystyle {\left(x^2\right)}^2}$.

Reorder the factors of ${\displaystyle \frac{5}{1+25x^4{y}^2}(x^{2}y\mathrm{<mo>\prime </mo>}+2yx)}$.

Multiply ${\displaystyle x^{2}y\mathrm{<mo>\prime </mo>}+2yx}$ by ${\displaystyle \frac{5}{1+25x^4{y}^2}}$.

Factor ${\displaystyle x}$ out of ${\displaystyle x^{2}y\mathrm{<mo>\prime </mo>}+2yx}$.

Move ${\displaystyle 5}$ to the left of ${\displaystyle x(xy\mathrm{<mo>\prime </mo>}+2y)}$.

By the Sum Rule, the derivative of ${\displaystyle x+4x{y}^2}$ with respect to ${\displaystyle x}$ is ${\displaystyle \frac{d}{dx}\left[x\right]+\frac{d}{dx}\left[4x{y}^2\right]}$.

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

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

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 ${\displaystyle g\left(x\right)=y^2}$.

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 ${\displaystyle f\left(x\right)=x^2}$ and ${\displaystyle g\left(x\right)=y}$.

Rewrite ${\displaystyle \frac{d}{dx}\left[y\right]}$ as ${\displaystyle y\mathrm{<mo>\prime </mo>}}$.

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

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

Multiply ${\displaystyle y^2}$ by ${\displaystyle 1}$.

Apply the distributive property.

Multiply ${\displaystyle 2}$ by ${\displaystyle 4}$.

Reorder terms.

Simplify the left side.

Simplify the right side.

Since ${\displaystyle y\mathrm{<mo>\prime </mo>}}$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

Subtract ${\displaystyle 5y\mathrm{<mo>\prime </mo>}{x}^2}$ from both sides of the equation.

Move all terms not containing ${\displaystyle y\mathrm{<mo>\prime </mo>}}$ to the right side of the equation.