How do you differentiate #1/cos(x) = x/y^2-y#?

Answer 1

Use implicit differentiation and algebra to get #dy/dx=(y(1-sec(x)tan(x)y^2) )/(2x+y^3)#

Assume that the given equation, which is equivalent to #sec(x)=x/y^2-y#, implicitly defines #y# as a function of #x# (the assumption is true wherever the graph of this equation does not have a vertical tangent line).
Now differentiate both sides with respect to #x#, keeping in mind the assumption we made while also using the quotient rule and chain rule:
#sec(x)tan(x)=(y^2-2xy * dy/dx)/y^4-dy/dx#.

This simplifies to

#sec(x)tan(x)=dy/dx(-(2x+y^3)/y^3)+1/y^2#
Now solve for #dy/dx# to get
#dy/dx=(sec(x)tan(x)-1/y^2)*(-y^3/(2x+y^3))#
#=(1-sec(x)tan(x)y^2)/y^2 * y^3/(2x+y^3)#
#=(y(1-sec(x)tan(x)y^2) )/(2x+y^3)#
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Answer 2

To differentiate the equation ( \frac{1}{\cos(x)} = \frac{x}{y^2 - y} ) with respect to (x), you would follow these steps:

  1. Differentiate both sides of the equation with respect to (x).
  2. Use the chain rule and the quotient rule where necessary.
  3. Solve for ( \frac{dy}{dx} ) or any other derivatives if required.

Let's go through the steps:

  1. Differentiate the left side: [ \frac{d}{dx} \left( \frac{1}{\cos(x)} \right) = -\frac{\sin(x)}{\cos^2(x)} ]

  2. Differentiate the right side: [ \frac{d}{dx} \left( \frac{x}{y^2 - y} \right) = \frac{(y^2 - y) \cdot \frac{d}{dx}(x) - x \cdot \frac{d}{dx}(y^2 - y)}{(y^2 - y)^2} ]

  3. Simplify and solve for ( \frac{dy}{dx} ) if needed.

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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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