I'm trying to calculate the derivative of this: P= mv/ sqrt(1-v^2/c^2)where m and c are constants, and v depends on t. I know about the chain and product rules but I'm not even sure where to start. :/ Thank you for your help.
p = mv * (1-v^2/c^2)^(-1/2)
p' = m * (1-v^2/c^2)^(-1/2) - mv * (-1/2) (1-v^2/c^2)^(-3/2) * (-2v/c^2)
= [m(1-v^2/c^2) + mv^2/c^2]/(1-v^2/c^2)^(-3/2)
= m/(1-v^2/c^2)&(3/2)
Can you please explain how you derived the first half of the second line before the '-mv * (...)'? Thank you.
product rule
p = f*g
p' = f'g + fg'
I'm still not quite sure how that works out, if you can explain it please? Isn't the derivative of v dv/dt and the derivative of m (since it's a constant) 0?
f = mv
f' = m
g = (1-v^2/c^2)^(-1/2)
using the chain rule, g = h^(-1/2), so g' = (-1/2) h^(-3/2) h'
g' = (-1/2) (1-v^2/c^2)^(-3/2) (-2v/c^2)
and the rest follows
To calculate the derivative of the function P = mv/√(1 - v^2/c^2), where m and c are constants and v depends on t, we can use the quotient rule in combination with the chain rule.
Let's start with the quotient rule, which states that for two functions u(t) and v(t):
(d/dt) [u(t)/v(t)] = [v(t) * u'(t) - u(t) * v'(t)] / [v(t)]^2
In this case, our numerator will be the product of v(t) and u'(t), and our denominator will be [v(t)]^2. Let's identify our u(t) and v(t):
u(t) = mv
v(t) = √(1 - v^2/c^2)
Now, we need to calculate the derivatives of u(t) and v(t) with respect to t. Starting with the easier one, the derivative of u(t) = mv with respect to t, we simply apply the product rule:
u'(t) = m * v' + v * m'
Next, let's find the derivative of v(t) = √(1 - v^2/c^2). This requires the chain rule since v is a function of t. Let's define the inner function g(v):
g(v) = √(1 - v^2/c^2)
Using the chain rule, we can find g'(v):
g'(v) = (1/2) * (1 - v^2/c^2)^(-1/2) * (-2v) * (-1/c^2)
= (v/c^2) / √(1 - v^2/c^2)
Now, to find v'(t) = (d/dt) [g(v)], we multiply g'(v) by the derivative of v with respect to t, which is dv/dt:
v'(t) = (v/c^2) / √(1 - v^2/c^2) * (dv/dt)
Now, we have all the pieces to find the derivative of P = mv/√(1 - v^2/c^2). Applying the quotient rule, we can write:
dP/dt = [v * (m * (v/c^2) / √(1 - v^2/c^2) * (dv/dt)) - mv * (v/c^2) / √(1 - v^2/c^2) * (dv/dt)] / [v]^2
Simplifying this expression further will involve some algebraic manipulation, but this is the general approach for finding the derivative of P with respect to t.