# Problem in applying the Chain Rule

• navneet9431
In summary: Also be careful with the notation, and use parentheses to indicate where the arguments of the functions are.
navneet9431
Gold Member

## Homework Statement

I am facing problem in applying the chain rule.
The question which I am trying to solve is,
" Find the second derivative of
"

## The Attempt at a Solution

So, differentiated it the first time,
[BY CHAIN RULE]

And now to find the second derivative I differentiated it once again,
so,

=>

But this is a wrong answer.
Please tell me where am I doing the mistake in applying the chain rule?
I will be thankful for help!

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Delta2
navneet9431 said:

## Homework Statement

I am facing problem in applying the chain rule.
The question which I am trying to solve is,
" Find the second derivative of View attachment 227541 "

## Homework Equations

View attachment 227542

## The Attempt at a Solution

So, differentiated it the first time,
View attachment 227543 [BY CHAIN RULE]
View attachment 227544
And now to find the second derivative I differentiated it once again,
so,
View attachment 227545
=>View attachment 227546
But this is a wrong answer.
Please tell me where am I doing the mistake in applying the chain rule?
I will be thankful for help!
You differentiated ##\dfrac{d}{dx}(fg)## to ##\dfrac{d}{dx}(f)\cdot \dfrac{d}{dx}(g)## which it is not, and it is not the chain rule.
Do you know the chain rule? I would have expected this information under point 2. of the template.

navneet9431 said:

## Homework Statement

I am facing problem in applying the chain rule.
The question which I am trying to solve is,
" Find the second derivative of View attachment 227541 "

## Homework Equations

View attachment 227542

## The Attempt at a Solution

So, differentiated it the first time,
View attachment 227543 [BY CHAIN RULE]
View attachment 227544
And now to find the second derivative I differentiated it once again,
so,
View attachment 227545
=>View attachment 227546
But this is a wrong answer.
Please tell me where am I doing the mistake in applying the chain rule?
I will be thankful for help!

To get ##d^2y/dt^2## you need to apply the product rule to ##dy/dt##. That will produce two terms, not one, although you can then simplify it down to one term again.

Please do NOT attach images; it makes it difficult to cite results and sub-results. Since you already used some kind of package to format your formulas, why not type them in here directly, using LaTeX?

Delta2
Your mistake is not in the application of the chain rule but you don't seem to apply correctly the product rule. You have find ##\frac{dy}{dt}## as product of ##t## and ##(t^2+1)^{-\frac{1}{2}}##. So to calculate the derivative of that product first apply correctly the product rule ##\frac{d(fg)}{dt}=\frac{df}{dt}g+f\frac{dg}{dt}## for ##f(t)=t## and ##g(t)=(t^2+1)^{-\frac{1}{2}}##and then apply the chain rule to calculate correctly ##\frac{dg}{dt}##.

navneet9431

## 1. What is the Chain Rule in calculus?

The Chain Rule is a rule in calculus that allows us to find the derivative of a composite function, where one function is inside another. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

## 2. Why is the Chain Rule important in mathematics?

The Chain Rule is important because it allows us to find the rate of change of complex functions made up of multiple simpler functions. It is a fundamental concept in calculus and is used extensively in many areas of mathematics, physics, and engineering.

## 3. What are some common mistakes made when applying the Chain Rule?

Some common mistakes made when applying the Chain Rule include forgetting to apply the derivative to the inner function, incorrectly differentiating the outer function, and not simplifying the final answer. It is also important to pay attention to the order of operations when using the Chain Rule.

## 4. How can the Chain Rule be applied to real-world problems?

The Chain Rule can be applied to real-world problems in fields such as physics, economics, and engineering. For example, it can be used to find the rate of change of a quantity in a system that is affected by multiple variables, or to optimize a function by finding its critical points.

## 5. Are there any alternative methods to the Chain Rule?

Yes, there are alternative methods to the Chain Rule such as the Power Rule, Product Rule, and Quotient Rule. These rules can be used in specific cases where the Chain Rule may not be applicable or may be more complicated to use. However, the Chain Rule is a general rule that can be applied to any composite function.

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