Question

if you x^3 y e^z where x = t , y= t^2 and z = in t find du/ dt at t = 2 , solve this question ?

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Answer 1

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Suppose

f (x, y) = xy sin(xy) .

We could find the partial derivatives in the usual way as

∂f∂x= y sin(xy) + xy 2

cos(xy)∂f∂x

= x sin(xy) + x2

y cos(xy)

Function f is said to be a composite function.

Notice that f is effectively a function of t alone

⇒ Total df dt exists.

We can write the following chain rule:

Chain Rule: When f = f (x, y) and x = x(t) and y = y(t):

ddt=∂f+∂x+dx+dt+∂f+∂y+dy+dt

$\large\mathbb\blue{☑MORE\:TO\:KNOW☑}$

The proof of this theorem uses the definition of differentiability of a function of two variables. Suppose that f is differentiable at the point P(x0,y0), where x0=g(t0) and y0=h(t0) for a fixed value of t0. We wish to prove that z=f(x(t),y(t)) is differentiable at t=t0 and that holds at that point as well.

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$\large\mathbb\green{HOPE\:IT\:HELPS}$

Answer 2

Step-by-step explanation:

Answer and solution is given in the image

Please see to it

Hope this is helpful