How do you implicitly differentiate #2= ysinx-xcosy #?

Answer 1

#y'=(cos y-y*cos x)/(sin x+x*sin y)#

Start with the given equation and differentiate with respect to x both sides of the equation

#2=y*sin x-x*cos y#
#d/dx(2)=d/dx(y*sin x-x*cos y)#
#0=y*d/dx(sin x)+sin x*d/dx(y)-[x*d/dx(cos y)+cos y*d/dx(x)]#
#0=y*cos x+sin x*y'-[x(-sin y*y')+cos y*1]#
#0=y*cos x+sin x*y'+x*sin y*y'-cos y#
transpose terms without the #y'#
#cos y-y*cos x=sin x*y'+x*sin y*y'#

by symmetric property of equality we have

#sin x*y'+x*sin y*y'=cos y-y*cos x#
factor out #y'#
#(sin x+x*sin y)*y'=cos y-y*cos x#
Divide both sides of the equation by #(sin x+x*sin y)#
#((sin x+x*sin y)*y')/(sin x+x*sin y)=(cos y-y*cos x)/(sin x+x*sin y)#
#(cancel(sin x+x*sin y)*y')/cancel(sin x+x*sin y)=(cos y-y*cos x)/(sin x+x*sin y)#
#y'=(cos y-y*cos x)/(sin x+x*sin y)#

God bless....I hope the explanation is useful.

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Answer 2

To implicitly differentiate (2 = y \sin(x) - x \cos(y)):

  1. Differentiate each term of the equation with respect to (x).
  2. Apply the chain rule where necessary.
  3. Solve for (\frac{{dy}}{{dx}}) to find the derivative of (y) with respect to (x).

Differentiating (2 = y \sin(x) - x \cos(y)) with respect to (x):

[ \frac{{d}}{{dx}}(2) = \frac{{d}}{{dx}}(y \sin(x)) - \frac{{d}}{{dx}}(x \cos(y)) ]

Simplify the derivatives:

[ 0 = y\cos(x) + \sin(x)\frac{{dy}}{{dx}} + \cos(y) - x\left(-\sin(y)\frac{{dy}}{{dx}}\right) ]

Rearrange the terms to isolate (\frac{{dy}}{{dx}}):

[ \sin(x)\frac{{dy}}{{dx}} + \sin(y)\frac{{dy}}{{dx}} = -y\cos(x) - \cos(y) + x\sin(y) ]

Factor out (\frac{{dy}}{{dx}}):

[ \frac{{dy}}{{dx}}(\sin(x) + \sin(y)) = -y\cos(x) - \cos(y) + x\sin(y) ]

Finally, solve for (\frac{{dy}}{{dx}}):

[ \frac{{dy}}{{dx}} = \frac{{-y\cos(x) - \cos(y) + x\sin(y)}}{{\sin(x) + \sin(y)}} ]

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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.

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.

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.

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.

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