So the correct answer isdy/dx = (-sin4x - 4siny) / (4y + 4cos4x)

In summary, when using implicit differentiation to find dy/dx, it is important to use the chain rule correctly and to remember that the derivative of sine is cosine, not 1/cosine. In this particular problem, the correct answer for dy/dx is (-4sin4x-4siny+4)/(4y+cosy).
  • #1
JackieAnne
7
0
Use implicit differentiation to find dy/dx.
y is a differentiable function of x

2y^2+4xsiny = cos4x

Here is what I did:

4y*dy/dx + 4siny+ 1/cosy*dy/dx = -sin4x + 4
4y*dy/dx + 1/cosy*dy/dx = -sin4x - 4siny + 4
dy/dx(4y + 1/cosy) = -sin4x - 4siny + 4
dy/dx = (-sin4x - 4siny + 4) / (4y + (1/cosy))

My answer is incorrect. Can anyone tell me where I went wrong and what the answer should be? Thanks
 
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  • #2
JackieAnne said:
2y^2+4xsiny = cos4x

Here is what I did:

4y*dy/dx + 4siny+ 1/cosy*dy/dx = -sin4x + 4

For the bold part on the left hand side, you want to use

[tex] \frac{d}{dy} \sin y = \cos y, [/tex]

not 1/cos y. For the bold part on the right hand side, use the chain rule to see that

[tex] \frac{d}{dx} \cos(4x) = - \sin(4x) \frac{d(4x)}{dx} = - 4\sin(4x) . [/tex]
 

What is a differentiable function?

A differentiable function is a mathematical function that is continuous and has a well-defined derivative at every point in its domain. This means that the function is smooth and has no abrupt changes or corners.

How do you determine if a function is differentiable?

A function is differentiable if it meets the following criteria: it is continuous at every point in its domain, it has a well-defined derivative at every point in its domain, and the derivative is equal to the slope of the tangent line at each point.

What is the difference between differentiable and non-differentiable functions?

The main difference between differentiable and non-differentiable functions is that differentiable functions have well-defined derivatives at every point in their domain, while non-differentiable functions do not. This means that non-differentiable functions have sharp changes or corners, making it impossible to find a consistent slope at every point.

Can a function be continuous but not differentiable?

Yes, it is possible for a function to be continuous but not differentiable. This occurs when the function has a sharp change or corner at a certain point, making it impossible to find a consistent slope for the entire function.

Why is differentiability important in calculus and real-world applications?

Differentiability is important in calculus because it allows us to calculate the rate of change of a function at any point, as well as approximate the behavior of the function near that point. In real-world applications, differentiable functions are used to model various phenomena, such as velocity, acceleration, and growth rates.

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