5. Find the equation of the tangent plane to x^2 + y^2 + z^2...

5. Find the equation of the tangent plane to $x^2 + y^2 + z^2 = 4$ at the point $(1, 1, \sqrt{2})$. 6. Let $f(x, y, z) = 6xy - z^2$, $x = 3r \cos \theta$, $y = \cos^2 \theta$, and $z = 7r$. Use the chain rule to calculate $\frac{\partial f}{\partial \theta}$. 7. Calculate the partial derivative $\frac{\partial w}{\partial z}$ using implicit differentiation of $x^3w + w^4 + wz^2 + 3yz = 0$.

Transcript

00:01 Hello students, let's discuss the question.

00:03 Here we have to use the chain rules to find the indicated partial derivatives.

00:08 So this is the function p and these are u, b and w in terms of x and y, and we have to find the partial derivative of p with respect to x and y.

00:22 When x is equal to 0 and y is equal to 3.

00:25 Now at x is equal to 0 and y is equal to 3, the values of u b and w are u is equal to 0, b is equal to 3 and w is equal to 1.

00:42 Now we will differentiate the del p divided by del x is equal to 1 upon 1 divided by under root u squared plus plus v square plus w squared multiplied by 2 u multiplied by del u divided by del x plus this will be half here 2 v del v divided by del x plus 2 w multiplied by del x so this will be equal to 1 divided by under root u square plus v squared v plus w squared multiplied by u del u divided by del x plus b del v divided by del x plus w divided by del x now we will partially differentiate these three u v and w with respect to x so del u divided by del x is equal to e to the power y del v divided by del y is equal to sorry this will be del x is equal to y multiplied by e to b power x and del w divided by del x is equal to y multiplied by e to b power x multiplied by y now at this point these values will be at x comma y is equal to 0 .3 we have del u divided by del x is equal to e to the power 3 del b divided by del x is equal to 3 and del w divided by del x is equal to 3 this is also 3 so proceeding ahead now del p divided by del x is equal to 3 at 0 .3, we have 1 divided by under root 0 squared plus 9 plus 1 multiplied by 0 multiplied by 0 multiplied by e to the power 3 plus 3 multiplied by 3 plus 3 multiplied by 1.

03:42 This gives 10 divided by under root 10 and we get under root 10.

03:49 Is the answer of the first part.

03:52 Now similarly, we'll find the value of del p divided by del x...

Question

Please give Ace some feedback