Differentiating the differences!

Calculus Level 1

If ${y=\sin(x)}$ , and $\dfrac{\text{d}(x)}{\text{d}(t)}=0$ what is $\dfrac{\text{d}(y)}{\text{d}(t)}$ ?

\sin(x)

x

-\cos(x)

0

\cos(x)

4 solutions

Steven Chase
Jul 18, 2020

$\frac{dy}{dt} = \frac{dy}{dx} \frac{dx}{dt} = \cos(x) \frac{dx}{dt} = 0$

I read it somewhere that when another variable pops out, which is of no relation to $y$ or $x$ as given, then the derivative is $0$ .

Vinayak Srivastava - 10 months, 4 weeks ago

Whether or not $x$ is related to $t$ depends on the context. Sometimes it does and sometimes it doesn't. Here, we have no context.

Steven Chase - 10 months, 4 weeks ago

So is this problem wrong? If yes, please tell me, I will delete it.

Vinayak Srivastava - 10 months, 4 weeks ago

@Vinayak Srivastava – It would help if you said that $\frac{dx}{dt} = 0$

Steven Chase - 10 months, 4 weeks ago

@Steven Chase – Edited. Thanks!

Vinayak Srivastava - 10 months, 4 weeks ago

@Vinayak Srivastava – Thanks, I have edited my solution as well

Steven Chase - 10 months, 4 weeks ago

@Steven Chase Excellent solution Upvoted!

A Former Brilliant Member - 10 months, 4 weeks ago

Siddharth Chakravarty
Jul 19, 2020

I solved this just by logic, the change of x with respect to t is 0, i.e nothing. So t does not affect x. And also as y=sin(x), y is affected by x but we found out that t does not affect x, so y should also be not affected t, thus change of y with respect to t would be 0.

Nice logic!

Vinayak Srivastava - 10 months, 4 weeks ago

Thank you.

Siddharth Chakravarty - 10 months, 4 weeks ago

Chew-Seong Cheong
Jul 19, 2020

$\begin{aligned} y & = \sin x \\ \implies \frac {dy}{dt} & = \cos x \cdot \frac {dx}{dt} = \cos x \cdot 0 = \boxed 0 \end{aligned}$

Sir, what is the meaning of the symbol that @Alak Bhattacharya Sir used in the comment below my solution? I don't understand the working. Thanks!

Vinayak Srivastava - 10 months, 4 weeks ago

@Vinayak Srivastava it's partial derivative. When you partial derivative a function w.r.t. a variable, all other variables are considered constant. So, it's basically differentiation with other variables constant. You'll need it in coordinate geometry in 11th, but not in much detail. :)

Aryan Sanghi - 10 months, 4 weeks ago

I haven't learnt Co-ordinate Geometry, but there is one chapter I saw in my book, I haven't read it :)

Vinayak Srivastava - 10 months, 4 weeks ago

@Vinayak Srivastava – Arre, I mean it is used in coordinate geometry, it is not taught in coordinate geometry. You'll learn it sooner or later as it is required to solve many questions in JEE maths.

Aryan Sanghi - 10 months, 4 weeks ago

@Aryan Sanghi – Oh, sorry, I did not understand. It is not in my class, so I won't disturb anyone asking about it. Sorry!

Vinayak Srivastava - 10 months, 4 weeks ago

It is just $\dfrac {dz}{dt} = \dfrac {\partial z}{\partial x} \cdot \dfrac {dx}{dt} + \dfrac {\partial z}{\partial y} \cdot \dfrac {dy}{dt} = \cos x \sin y \cdot \dfrac {dx}{dt} + \sin x \cos y \cdot \dfrac {dy}{dt}$ .

Chew-Seong Cheong - 10 months, 4 weeks ago

What is the symbol used after the equal sign before x and z?

Vinayak Srivastava - 10 months, 4 weeks ago

Partial differentiation \partial (in LaTex). We used it when there is more than one variable. Because $z(x,y)$ .

Chew-Seong Cheong - 10 months, 4 weeks ago

@Chew-Seong Cheong – I don't have to study Calculus for Math till 11th, I am studying the basics for Physics problems. Sir, is partial differentiation important for Physics? Do I need to study it?

Vinayak Srivastava - 10 months, 4 weeks ago

@Vinayak Srivastava – Yes, it is important to Physics partial differentiation was invented to due with more than one dimensions such as fluid mechanics (pressure, flow rate, density) and electromagnetism (3 dimensional space). But no worry you will learn sooner or later. It works just like $\frac d{dx}$ , just keep other variables as constants. For example, $z=\sin x \sin y$ , $\frac {\partial z}{\partial x} = \cos x \sin y$ and $\frac {\partial z}{\partial y} = \sin x \cos y$ . As simple as that.

Chew-Seong Cheong - 10 months, 4 weeks ago

@Chew-Seong Cheong – I think it should come later in my book, thank you for replying!

Vinayak Srivastava - 10 months, 4 weeks ago

@Vinayak Srivastava – Nope! Only in university

Chew-Seong Cheong - 10 months, 4 weeks ago

@Chew-Seong Cheong – Oh, then I don't think it will be asked in my exam.

Vinayak Srivastava - 10 months, 4 weeks ago

Vinayak Srivastava
Jul 18, 2020

${y=\sin(x)}$ , $\dfrac{\text{d}(x)}{\text{d}(t)}=0$

$\dfrac{\text{d}(y)}{\text{d}(t)}=\dfrac{\text{d}(y)}{\text{d}(x)} \dfrac{\text{d}(x)}{\text{d}(t)}=\cos(x)\times 0 =\boxed{0}$

Now try to calculate $\dfrac {dz}{dt}$ if $z=\sin x\sin y$ . Then generalize the case for $z=f(x,y)$ .

A Former Brilliant Member - 10 months, 4 weeks ago

I don't know how to do it. Can you tell me, Sir?

Vinayak Srivastava - 10 months, 4 weeks ago

If $z=f(x,y)$ , then $\dfrac {dz}{dt}=\dfrac {\partial z}{\partial x}.\dfrac {dx}{dt}+\dfrac {\partial z}{\partial y}.\dfrac {dy}{dt}$

For $z=\sin x\sin y,\dfrac {dz}{dt}=\cos x\sin y\dfrac {dx}{dt}+\sin x\cos y\dfrac{dy}{dt}$ .

A Former Brilliant Member - 10 months, 4 weeks ago

@A Former Brilliant Member – What is the meaning of the symbol? Sorry, I don't understand.

Vinayak Srivastava - 10 months, 4 weeks ago

@Vinayak Srivastava – It is the symbol for partial derivative, that is, the derivative of a function with respect to one of it's arguments, keeping the other arguments constant. See the example I have given to you, namely the function $z=\sin x\cos x$ .

A Former Brilliant Member - 10 months, 4 weeks ago

Sir, can you please help me with this problem ?

Vinayak Srivastava - 10 months, 4 weeks ago

To solve that problem, you have to know modular arithmetic, and Chinese Remainder Theorem . Study about them.

A Former Brilliant Member - 10 months, 4 weeks ago

Differentiating the differences!

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