2022-01-04

Find the directional derivative of at a given point in the direction indicated by the angle theta. F (x,y)=x^3y^4+x^4y^3, (1,1)

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$F\left(x,y\right)={x}^{3}{y}^{4}+{x}^{4}{y}^{3}$

By the Sum Rule, the derivative of ${x}^{3}{y}^{4}+{x}^{4}{y}^{3}$ with respect to $x$ is

$\frac{d}{dx}\left[{x}^{3}{y}^{4}\right]+\frac{d}{dx}\left[{x}^{4}{y}^{3}\right]$

Evaluate $\frac{d}{dx}\left[{x}^{3}{y}^{4}\right]$

Since is constant with respect to $x$, the derivative of ${x}^{3}{y}^{4}$ with respect to $x$ is ${y}^{4}\frac{d}{dx}\left[{x}^{3}\right]$.

${y}^{4}\frac{d}{dx}\left[{x}^{3}\right]+\frac{d}{dx}\left[{x}^{4}{y}^{3}\right]$

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

${y}^{4}\left(3{x}^{2}\right)+\frac{d}{dx}\left[{x}^{4}{y}^{3}\right]$

Move $3$  to the left of

$3{y}^{4}{x}^{2}+\frac{d}{dx}\left[{x}^{4}{y}^{3}\right]$

Evaluate $\frac{d}{dx}\left[{x}^{4}{y}^{3}\right]$

Since ${y}^{3}$ is constant with respect to $x$, the derivative of ${x}^{4}{y}^{3}$ with respect to $x$ is ${y}^{3}\frac{d}{dx}\left[{x}^{4}\right]$

$3{y}^{4}{x}^{2}+{y}^{3}\frac{d}{dx}\left[{x}^{4}\right]$

Differentiate using the Power Rule which states that $\frac{d}{dx}\left[{x}^{n}\right]$ is $n{x}^{n-1}$ where $n=4$

$3{y}^{4}{x}^{2}+{y}^{3}\left(4{x}^{3}\right)$

Move $4$ to the left of ${y}^{3}$

$3{y}^{4}{x}^{2}+4{y}^{3}{x}^{3}$

Then put $\left(1,1\right)$ into derivative, where

$x=1$ and $y=1$

$3×{1}^{4}×{1}^{2}+4×{1}^{3}×{1}^{3}$

$=3+4$

$=7$ - Answer

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