Air conditionner
New member
- Joined
- Jun 29, 2021
- Messages
- 24
It is not necessary to "separate" any variable to differentiate. Is the 't' independent variable? What is the dependent variable? "What" you have posted is not an equation - it does NOT have "equal" to ( = ) sign.Hi,
I have stumbled upon a little equation: I can't differentiate it.
Could anyon help me?
View attachment 28178
It's this one. I know how to deal with the products, and transform the root in fractional exponents, but I can't separate the t from the rest of the expression.
Thanks in advance,
Air conditioner
This is a function of t, so presumably you want to differentiate it with respect to t. I don't know what you mean by "separate the t from the rest of the expression".Hi,
I have stumbled upon a little equation: I can't differentiate it.
Could anyon help me?
View attachment 28178
It's this one. I know how to deal with the products, and transform the root in fractional exponents, but I can't separate the t from the rest of the expression.
Thanks in advance,
Air conditioner
Do you know the chain rule? If `u = t - 125`, then this is `1/10 u^{1/3}`, which you should be able to differentiate.Well, this is the exercise: View attachment 28179
I want to find the derivative of y with respect to t. I know that 0.5 disappears since it's a constant. And after that, I can get this:
View attachment 28180
But I can't go further... I don't know how to expand with fractional exponents. Should I use the binomial theorem? I used an application (Gauthmath) to try to get an answer, and they did give a solution (where they didn't use the binomial theorem) but I didn't understand the steps...
\(\displaystyle y \ = \ 0.5 \ + \ \sqrt[3]{t \ - \ 125} \).............. substitute \(\displaystyle u \ = \ -125 \ + \ t \).........→...... \(\displaystyle \frac{du}{dt} \ = 1 \)Well, this is the exercise: View attachment 28179
I want to find the derivative of y with respect to t. I know that 0.5 disappears since it's a constant. And after that, I can get this:
View attachment 28180
But I can't go further... I don't know how to expand with fractional exponents. Should I use the binomial theorem? I used an application (Gauthmath) to try to get an answer, and they did give a solution (where they didn't use the binomial theorem) but I didn't understand the steps...
Thank you.\(\displaystyle y \ = \ 0.5 \ + \ \sqrt[3]{t \ - \ 125} \).............. substitute \(\displaystyle u \ = \ -125 \ + \ t \).........→...... \(\displaystyle \frac{du}{dt} \ = 1 \)
\(\displaystyle y \ = \ 0.5 \ + \ \sqrt[3]{u} \ = \ 0.5 + u^{\frac{1}{3}}\)
\(\displaystyle \frac{dy}{dt} \ = \frac{dy}{du} \ * \ \frac{du}{dt}\) .................. apply chain rule
\(\displaystyle \frac{dy}{dt} \ = \left[ \frac{1}{3}u^{(\frac{1}{3}-1)}\right] \ * \ \frac{du}{dt}\)
\(\displaystyle \frac{dy}{dt} \ = \left[ \frac{1}{3}u^{(-\frac{2}{3})}\right] \ * \ (1) \)
\(\displaystyle \frac{dy}{dt} \ = \ \frac{1}{3 \ * u^{(\frac{2}{3})}} \) ............. Now substitute back for "u".
Okay!Study that carefully- you are going to see functions that are a lot harder to diffentiate!