# Take the derivative of an equation

#### atty

##### New member
Hi,

I'm trying to learn electronics and math in my spare time, and now I'm stuck somewhere
with electronics because my math isn't quite that good yet.

My question is how do they get to the 3. equation, I think you have to take the derivative of the 2. equation and then add it into the 1. equation ?

Can anyone help?

#### Subhotosh Khan

##### Super Moderator
Staff member
Hi,

I'm trying to learn electronics and math in my spare time, and now I'm stuck somewhere
with electronics because my math isn't quite that good yet.

View attachment 28281

My question is how do they get to the 3. equation, I think you have to take the derivative of the 2. equation and then add it into the 1. equation ?

Can anyone help?

Please try the "action" as you have suggested (described) and show us the result.

If you have trouble, please tell us exactly where did you get stuck - we will unstuck you.

#### HallsofIvy

##### Elite Member
Hi,

I'm trying to learn electronics and math in my spare time, and now I'm stuck somewhere
with electronics because my math isn't quite that good yet.

View attachment 28281

You are given that $$\displaystyle L\frac{di(t)a}{dt}+ R_1i(t)+ v(t)= 0$$ and that $$\displaystyle i(t)= C\frac{dv(t)}{dt}+ \frac{v(t)}{R_2}$$ and are told to replace i(t) in the first equation by its form in the second equation. That will give you $$\displaystyle L\frac{d(C\frac{dv(t)}{dt}+ \frac{v(t)}{R_2})}{dt}+ R_1i(t)+ v(t)= 0$$. Yes, you now have to differentiate that expression- and since that expression already has a derivative the result will have a second derivative. That will give $$\displaystyle LC\frac{d^2v(t)}{dt^2}+\frac{1}{R_2}\frac{dv(t)}{dt}+ R_1C\frac{dv(t)}{dt}+ \frac{R_1}{R_2}v(t)+ v(t)= 0$$.

#### atty

##### New member
Please try the "action" as you have suggested (described) and show us the result.

If you have trouble, please tell us exactly where did you get stuck - we will unstuck you.

I would like to calculate the derivative of the 2nd equation and then add this result with the 1st equation to get to the 3rd equation, but I don't know how to get started.

#### atty

##### New member
Okay, but could you show me how to calculate the derivative of i(t)?

#### Eugenio

##### New member
Differentiating equation 2, we get:
$\frac{\mathrm{d} i}{\mathrm{d} t}=c\frac{d^2v}{dt^2}+\frac{1}{R_2}\frac{dv}{dt}$
Replacing in 1, leads to:$L\left [C\frac{d^2v}{dt^2}+\frac{1}{R_2}\frac{dv}{dt}) \right ]+R_1\left [C\frac{dv}{dt}+\frac{1}{R_2}v(t)) \right ]+v(t)=0$

#### atty

##### New member
Differentiating equation 2, we get:
$\frac{\mathrm{d} i}{\mathrm{d} t}=c\frac{d^2v}{dt^2}+\frac{1}{R_2}\frac{dv}{dt}$
Replacing in 1, leads to:$L\left [C\frac{d^2v}{dt^2}+\frac{1}{R_2}\frac{dv}{dt}) \right ]+R_1\left [C\frac{dv}{dt}+\frac{1}{R_2}v(t)) \right ]+v(t)=0$

Thanks for helping.