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How do you differentiate #cos(x/y) = x/y#?

Answer 1

Isabella Ackerman

#(dy)/(dx) = 1/t_1# where #t_1# is the unique root of #cos(t)=t#

The equation #cos(t) = t# has only one solution, #t_1 ~~ 0.7390851332#

So our original equation amounts to #x/y = t_1#

Hence #y = x/t_1#

So #(dy)/(dx) = 1/t_1 ~~ 1.353024#

graph{(y-cos(x))(y-x)=0 [-4.44, 5.56, -2.03, 2.97]}

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Answer 2

Charles Anctil

To differentiate cos(x/y) = x/y, we will use the chain rule. The derivative of cos(u) with respect to u is -sin(u), and the derivative of u = x/y with respect to x is 1/y, and with respect to y is -x/y^2. Therefore, applying the chain rule, the derivative of cos(x/y) with respect to x is:

sin(x/y) * (1/y)

And the derivative with respect to y is:

sin(x/y) * (-x/y^2)

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.