How do you properly apply the chain rule in implicit differentiation?

In summary, the conversation discusses using Latex to type formulas on a website, but the person is having trouble with the formatting. They also mention a solution to a problem involving the chain rule in differentiation.
  • #1
Homework Statement
Find the implicit differentiation
Relevant Equations
(sinx)^(cosy)+(cosx)^(siny)=a
The working I've tried is in the attachment.
 

Attachments

  • 15980283088327760621644651835670.jpg
    15980283088327760621644651835670.jpg
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  • #3
I'm not going to follow your work, too taxing: I get this solution

$$\tfrac{dy}{dx}=\tfrac{( \cos x)^{\sin y}\sin y \tan x - ( \sin x)^{\cos y}\cos y \cot x}{( \cos x )^{ \sin y } \cos y \log \cos x - ( \sin x) ^{\cos y} \sin y \log \sin x}$$
 
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  • #4
fresh_42 said:
I guess people do not want to download your picture, then rotate it, zoom in, only to find out that they cannot read it anyway.

Here is how to type formulas on PF (it's not difficult):
https://www.physicsforums.com/help/latexhelp/
Ahh, I apologise.
I've tried using Latex as you have asked, but I'm afraid it's taking way too long to type out my working.
However, I've taken a better photo, I'm not confident in my working but please do check. Thank you :)
 

Attachments

  • Implicit Differentiation.pdf
    262.5 KB · Views: 145
  • #5
benorin said:
I'm not going to follow your work, too taxing: I get this solution

$$\tfrac{dy}{dx}=\tfrac{( \cos x)^{\sin y}\sin y \tan x - ( \sin x)^{\cos y}\cos y \cot x}{( \cos x )^{ \sin y } \cos y \log \cos x - ( \sin x) ^{\cos y} \sin y \log \sin x}$$
Thank you for the answer. But may I ask what working you've done to solve this?
 
  • #6
You forgot the chain rule when differentiating functions of y you need to multiply by y' from the chain rule, that'll give an equation involving y', solve it.
 
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Likes Blurry__face14

1. How do I identify when to use the chain rule in implicit differentiation?

The chain rule should be used when the function being differentiated is composed of two or more functions, where one function is nested within the other. This can be identified by looking for expressions such as f(g(x)) or g(f(x)).

2. What is the formula for applying the chain rule in implicit differentiation?

The formula for applying the chain rule in implicit differentiation is d/dx(f(g(x))) = f'(g(x)) * g'(x). This means that the derivative of the outer function is multiplied by the derivative of the inner function.

3. Can the chain rule be applied to any function in implicit differentiation?

Yes, the chain rule can be applied to any function that is composed of two or more nested functions. This includes polynomial, exponential, logarithmic, and trigonometric functions.

4. How do I handle constants when using the chain rule in implicit differentiation?

When differentiating a function with a constant, the constant can be treated as a coefficient and can be factored out of the derivative. For example, if the function is y = 3x^2, the derivative would be dy/dx = 6x.

5. Are there any common mistakes to avoid when applying the chain rule in implicit differentiation?

One common mistake is forgetting to apply the chain rule to both the inner and outer functions. It is important to differentiate each function separately and then multiply them together. Another mistake is not properly simplifying the final expression after applying the chain rule.

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