Understanding a quote about implicit differentiation

In summary, the conversation discusses the derivative of the inverse tangent function, which can be calculated using implicit differentiation. The book provides an example of this process, using Leibniz's notation. The resulting derivative is 1/(1+x^2). One person expresses confusion about the notation, but ultimately agrees that it is correct.
  • #1

mcastillo356

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I've got a solved calculation of the inverse tangent function by implicit differentiation I'm trying to understand
Hi PF

A personal translation of a quote from Spanish "Calculus", by Robert A. Adams:
Inverse tangent function derivative can be also obtained by implicit differentiation: if y=tan−1⁡x, then x=tan⁡y, and 1=(sec2⁡y)dydx=(1+tan2⁡y)dydx=(1+x2)dydx Hence, ddxtan−1⁡x=11+x2
It's about advice on Lebniz's notation1=(sec2⁡y)dydx means dxdx=(sec2⁡y)dydx, I'm quite sure. Why (sec2⁡y)dydx=(1+tan2⁡y)dydx? But I'm also quite sure that the right notation for (sec2⁡y)dydx=(1+tan2⁡y)dydx would be (sec2⁡y)ddx=(1+tan2⁡y)ddx
 
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  • #2
##sec^2y = 1 + tan^2 y## is one of the most important trig identities.
 
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  • #3
Sorry, I've posted nonsense: I was trying to be clever about Leibniz's notation: impose my personal point of view. I've got troubles posting and editing. My weird opinion was... Well, I will post again, and then my unfounded opinion:

Quote from the book:

The derivative of the inverse tangent function can be calculated also by implicit differentiation: if ##y=\tan^{-1} x##, then ##x=\tan y##, and

$$1=(\sec^2 y)\dfrac{dy}{dx}=(1+\tan^2 y)\dfrac{dy}{dx}=(1+x^2)\dfrac{dy}{dx}$$

Hence

$$\dfrac{d}{dx}\tan^{-1}x=\dfrac{1}{1+x^2}$$

My botched job: set notations like ##(\sec^2 y)\dfrac{d}{dx}##

I am not native. Forgive my English.

Greetings!
 
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  • #4
Sorry, I've posted nonsense: I was trying to be clever about Leibniz's notation: impose my personal point of view. I've got troubles posting and editing. My weird opinion was... Well, I will post again, and then my unfounded opinion:

Quote from the book:

The derivative of the inverse tangent function can be calculated also by implicit differentiation: if ##y=\tan^{-1} x##, then ##x=\tan y##, and

$$1=(\sec^2 y)\dfrac{dy}{dx}=(1+\tan^2 y)\dfrac{dy}{dx}=(1+x^2)\dfrac{dy}{dx}$$

Hence

$$\dfrac{d}{dx}\tan^{-1}x=\dfrac{1}{1+x^2}$$

My botched job: set notations like ##(\sec^2 y)\dfrac{d}{dx}##

I am not native. Forgive my English.

Greetings!
 
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  • #5
What's the problem with that? It all looks good to me.
 
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1. What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly defined in terms of its independent variable. It is used when a function cannot be easily solved for the independent variable.

2. What is the purpose of using implicit differentiation?

The purpose of implicit differentiation is to find the rate of change of a function at a specific point, which is represented by the derivative. This allows us to analyze the behavior of the function and make predictions about its future values.

3. How is implicit differentiation different from explicit differentiation?

Explicit differentiation is used to find the derivative of a function that is explicitly defined in terms of its independent variable. It involves using the power rule, product rule, quotient rule, and chain rule. On the other hand, implicit differentiation involves using the chain rule and treating the dependent variable as a function of the independent variable.

4. When should implicit differentiation be used?

Implicit differentiation should be used when a function cannot be easily solved for the independent variable. This often occurs when the function is in the form of an equation with both the dependent and independent variables present, making it difficult to isolate the independent variable.

5. What are some common applications of implicit differentiation?

Implicit differentiation is commonly used in physics, engineering, and economics to analyze rates of change and make predictions about the behavior of a system. It is also used in optimization problems to find the maximum or minimum value of a function.

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