Implicit differentiation function of other function.

Question

I have the following implicit function:

$$\frac{dS_k}{dt}= \frac{dS_k}{dS_i}*\frac{dS_i}{dt} $$

Now we have to derive on both sides of the equality by (applying the rule of the chain): $$\frac{d}{dt}$$

Obtaining the following result: $$\frac{d^2S_k}{dt^2}= \frac{dS_k}{dS_i}*\frac{d^2S_i}{dt^2}+\frac{d^2S_k}{dS_i^2}*(\frac{dS_i}{dt})^2 $$ The problem is, i can not get the same result: My "solution"

At this point it is impossible (for me) to arrive at an exact solution, as the book shows:Book solution and explanation

What am I doing wrong?.

Can someone solve it step by step ?. Please.

P.D: The equation is used for the numerical calculation of mechanisms.

Answer 1

Are you understanding what you are doing ?

In first line you write $\frac{d}{dt}$. If you don't write on the right side what function to differentiate, this means anything or nothing.

In second line you wrote $\frac{d^2 S_h}{dt}$ which is nonsense. Is it a typo ? Do you mean $\frac{d^2 S_h}{dt^2}$ ? . Anyway, the equation is not correct. See below.

Note : writing $\frac{d^2 S_h}{dt}$ means that $S_h$ is differentiated two times because $d^2$ at numerator. The first differentiation is with respect to $t$ because $dt$ at denominaror. The second differentiation is with respect to what ? This is missing at denominator.

$$\frac{d^2 S_k}{dt^2}=\frac{d \left(\frac{d S_k}{dt} \right)}{dt}$$

$$\frac{d S_k}{dt}=\frac{d S_k}{dS_i}\frac{d S_i}{dt}$$

$$\frac{d^2 S_k}{dt^2}=\frac{d (uv) }{dt}=u\frac{dv}{dt}+v\frac{d u}{dt} \quad \begin{cases}u=\frac{d S_k}{d S_i}\\v=\frac{d S_i}{dt} \end{cases}$$

$\frac{dv}{dt}=\frac{d \left(\frac{d S_i}{dt} \right)}{dt}=\frac{d^2 S_i}{dt^{\:2}}$

$\frac{du}{dt}=\frac{d \left(\frac{d S_k}{dS_i} \right)}{dt}=\frac{d \left(\frac{d S_k}{dS_i} \right)}{dS_i}\frac{d S_i}{dt}=\frac{d^2S_k}{dS_i^{\:2}}\frac{d S_i}{dt}$

$$\frac{d^2 S_k}{dt^2}=u\frac{dv}{dt}+v\frac{d u}{dt} = \frac{d S_k}{d S_i}\frac{d^2 S_i}{dt^{\:2}} +\frac{d S_i}{dt}\frac{d^2S_k}{dS_i^{\:2}}\frac{d S_i}{dt} $$

$$\frac{d^2 S_k}{dt^2}= \frac{d S_k}{d S_i}\frac{d^2 S_i}{dt^{\:2}} +\frac{d^2S_k}{dS_i^{\:2}}\left(\frac{d S_i}{dt}\right)^2 $$

Answer 2

So use the bracket, to indicate that the $\frac{d}{dt}$should be applied to both sides of the equality, I did not copy another $\frac{d}{dt}$ on the left, of course that is not done and it is wrong to do it, but i did it for laziness.

It is not a typographical error, it is how was solving it (I have to go back to review my classes and derivative books again). I do not know that was thinking in putting $\frac{d^2}{dt}$. I was not using logic, wanted to solve without understanding that was doing.

My big mistake is, that it separated the numerator and denominator, at the moment of derive, as if it were a simple fraction, for example $\frac{x}{y}$, with this reason remained a subtraction in the parenthesis.

Thank you @JJacquelin for help me. Now I can continue to understand and studying, this numerical method.