Name: Partial Derivatives
Uploaded: 2019-07-19
Duration: 1 h 33 s
Description: This calculus video explains how to find first-order partial derivatives of functions with two and three variables. This video contains plenty of examples an...

Question

If

z (x, y) = {xtan}^{- 1} (xy), x = t^{2}, y = {se}^{t}, s, t \in R

, Find

\frac{\partial z}{\partial s}

and

\frac{\partial z}{\partial t}

at

s = t = 1

.

Accepted Answer

Given, $z (x, y) = {xtan}^{- 1} (xy), x = t^{2}, y = {se}^{t}$

$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s}$

$\Rightarrow \frac{\partial z}{\partial s} = [\frac{xy}{1 + (xy)^{2}} + \tan^{- 1} (xy)] (0) + \frac{x^{2}}{1 + (xy)^{2}} (e^{t})$

$\Rightarrow \frac{\partial z}{\partial s} = \frac{x^{2} e^{t}}{1 + (xy)^{2}}$

Replace $(x, y)$ values,

$\frac{\partial z}{\partial s} = \frac{e^{t} \cdot t^{4}}{1 + (t^{2} {se}^{t})^{2}}$

At $s = t = 1$ , $s = t = 1 \Rightarrow x = 1, y = e$

$\frac{\partial z}{\partial s} = \frac{e}{1 + e^{2}}$

Now, $\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t}$

$\Rightarrow \frac{\partial z}{\partial t} = [\frac{xy}{1 + x^{2} y^{2}} + \tan^{- 1} (xy)] (2 t) + [\frac{x^{2}}{1 + x^{2} y^{2}}] ({se}^{t})$

At $s = t = 1 \Rightarrow x = 1, y = e$

$∴ \frac{\partial z}{\partial t} = \frac{3 e}{1 + e^{2}} + 2 \tan^{- 1} (e)$

If $z (x, y) = {xtan}^{- 1} (xy), x = t^{2}, y = {se}^{t}, s, t \in R$ , Find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$ at $s = t = 1$ .

Important Questions on Differentials and Partial Derivatives

Learn and Prepare for any exam you want !

Ifz(x, y)=xtan-1xy, x=t2, y=set, s, t∈R , Find ∂z∂s and ∂z∂t at s=t=1.

Important Questions on Differentials and Partial Derivatives

Learn and Prepare for any exam you want !

If $z (x, y) = {xtan}^{- 1} (xy), x = t^{2}, y = {se}^{t}, s, t \in R$ , Find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$ at $s = t = 1$ .