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Viktor [21]
1 year ago
12

Suppose that the derivable functions x=x(t) and y=y(t) satisfy xcosy=2.

Mathematics
1 answer:
ololo11 [35]1 year ago
4 0

Applying implicit differentiation, it is found that dy/dt when y=π/4 is of:

a-) -√2 / 2.

<h3>What is implicit differentiation?</h3>

Implicit differentiation is when we find the derivative of a function relative to a variable that is not in the definition of the function.

In this problem, the function is:

xcos(y) = 2.

The derivative is relative to t, applying the product rule, as follows:

\cos{y}\frac{dx}{dt} - x\sin{y}\frac{dy}{dt} = 0

\frac{dy}{dt} = \frac{\cos{y}\frac{dx}{dt}}{x\sin{y}}

Since dx/dt=−2, we have that:

\frac{dy}{dt} = -2\frac{\cos{y}}{x\sin{y}}

When y = π/4, x is given by:

xcos(y) = 2.

x = \frac{2}{\cos{\frac{\pi}{4}}} = \frac{2}{\frac{\sqrt{2}}{2}} = \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = 2\sqrt{2}

Hence:

\frac{dy}{dt} = -2\frac{\cos{y}}{x\sin{y}}

\frac{dy}{dt} = -\frac{1}{\sqrt{2}}\cot{y}

Since cot(pi/4) = 1, we have that:

\frac{dy}{dt} = -\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -\frac{\sqrt{2}}{2}

Which means that option a is correct.

More can be learned about implicit differentiation at brainly.com/question/25608353

#SPJ1

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