# Let z=e^(x^2+3y)+x^3y^2, where x=tcos r and y = rt^4. Use the chain rule fot multivariable functions (Cals III Chain Rule) to find (delz)/(delr) and (delz)/delt). Give your answers in terms of r and t only. Be sure to show all of your work.

Question
Multivariable functions
Let $$\displaystyle{z}={e}^{{{x}^{{2}}+{3}{y}}}+{x}^{{3}}{y}^{{2}}$$, where $$\displaystyle{x}={t}{\cos{{r}}}{\quad\text{and}\quad}{y}={r}{t}^{{4}}$$. Use the chain rule fot multivariable functions (Cals III Chain Rule) to find $$\displaystyle\frac{{\partial{z}}}{{\partial{r}}}$$ and $$\displaystyle\frac{{\partial{z}}}{\partial}{t}{)}$$. Give your answers in terms of r and t only. Be sure to show all of your work.

2021-01-06
z=e^(x^2+3y)+x^3y^2 (1) where x=tcos r and y = rt^4 Now use the chain rule for multivariable function then (delz)/(delr)=(delz)/(delx)xx(delx)/(delr)+(delz)/(dely)xx(dely)/delr) (delz)/(delr)=(del(e^(x^2+3y)+x^3y^3))/(delx)xx(del(t cos r))/(delr)+(del(e^(x^2+3y)+x^3y^3))/(dely)xx(del(rt^4))/(delr) (delz)/(delr)=(e^(x^2+3y)*2x+3x^2y^2)xx(-t sin r)+(e^(x^2+3y)*2x+3x^2y^2)xxt^4 (delz)/(delr)=-t sin r(2xe^(x^2+3y)*2x+3x^2y^2)+t^4(3e^(x^2+3y)*2x+3x^3y) Now put x=tcosr, y=rt^4 then (delz)/(delr)=-t sin r(2tcos r*e^((t cos r)^2+3rt^4)+3(t cos r)^2(rt^4)^2)+t^4(3e^((t cos r)^2+3rt^4)+2(t cos r)^3rt^4) (delz)/(delr)=-(2t^2sinrcosr * r^((t cos r)^2+3rt^4)+3r^2t^(11)sin r cos^2r)+(3t^4e^((t cos r)^2+3rt^4)+2rt^(11)cos^3r) (delz)/(delr)=^((t cos r)^2+3rt^4)(-t^2sin 2r+3t^4)-3r^2t^11sin r cos^2 r + 2rt^(11) cos^3r [because, sin2x=2sinxcosx]

### Relevant Questions

Let $$\displaystyle{z}{\left({x},{y}\right)}={e}^{{{3}{x}{y}}},{x}{\left({p},{q}\right)}=\frac{{p}}{{q}}{\quad\text{and}\quad}{y}{\left({p},{q}\right)}=\frac{{q}}{{p}}$$ are functions. Use multivariable chain rule of partial derivatives to find
(i) $$\displaystyle\frac{{\partial{z}}}{{\partial{p}}}$$
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(a)
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$$\displaystyleΣ{y}=$$
$$\displaystyleΣ{x}^{2}=$$
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$$\displaystyle{t}=$$
critical $$\displaystyle{t}=$$
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