What is the Implicit Differentiation Equation for eysinx=x+xy?

In summary, taking the derivative of both sides of the equation eysinx=x+xy results in ey*cosx + ey*sinx*y'=1+y+xy'. This equation is difficult to solve, and the question arises whether the problem could be solved if it were in the form A+ By'= C+ Dy'. In that case, y' could be expressed as a function of x and y.
  • #1
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Equation: eysinx=x+xy

I took the derivative of both sides.
For the side with eysinx, I used the product rule and chain rule to get: ey*cosx + ey*sinx*y'
For the side with x+xy, I used the sum and product rule to get 1+y+xy'

So my resulting equation is: ey*cosx + ey*sinx*y'=1+y+xy', which I don't know how to solve and is probably wrong.
 
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  • #2
Why would you not knowhow to solve that for y'? If the problem were A+ By'= C+ Dy', could you solve for y'?
 
  • #3
express y' as a function of x and y
 

FAQ: What is the Implicit Differentiation Equation for eysinx=x+xy?

What is implicit differentiation?

Implicit differentiation is a mathematical method used to find the derivative of an equation that is not explicitly in the form of y = f(x). It is commonly used when an equation contains both x and y variables and cannot be easily solved for y in terms of x.

Why is implicit differentiation useful?

Implicit differentiation allows us to find the derivative of a function without having to solve for y explicitly. This is helpful when the equation is too complex or difficult to solve algebraically, or when the equation is given in a more general form.

How is implicit differentiation different from explicit differentiation?

Explicit differentiation involves finding the derivative of an equation with respect to a specific variable, usually x. Implicit differentiation, on the other hand, finds the derivative of an equation with respect to both x and y, treating y as a function of x.

Can implicit differentiation be applied to any equation?

Yes, implicit differentiation can be applied to any equation that contains both x and y variables. However, for implicit differentiation to work, the equation must be differentiable and have a well-defined derivative.

What are some practical applications of implicit differentiation?

Implicit differentiation has many practical applications in fields such as physics, engineering, and economics. It can be used to find the slope of a curve, calculate rates of change, and solve optimization problems. It is also used in computer graphics and image processing to calculate gradients and find tangent lines to curves.

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