Suppose r =
4t
3−27
3
and
s = t
2 + 3t, determine ds
dr
To find the derivative of s with respect to r, we use the chain rule:
ds/dr = ds/dt * dt/dr
First, let's find ds/dt:
ds/dt = d/dt (t^2 + 3t) = 2t + 3
Now, let's find dt/dr:
r = 4t^3 - 27
dr/dt = 12t^2
dt/dr = 1 / (dr/dt)
dt/dr = 1 / (12t^2)
Putting it all together:
ds/dr = (2t + 3) * (1 / (12t^2))
ds/dr = (1 / 6t^2) + (1 / 4t)
Therefore, the derivative of s with respect to r is ds/dr = (1 / 6t^2) + (1 / 4t).
To determine ds/dr, we need to find the derivative of s with respect to r.
First, let's find the derivative of s with respect to t:
ds/dt = d/dt (t^2 + 3t)
To find the derivative of t^2, we use the power rule: d/dt (t^2) = 2t
To find the derivative of 3t, we use the constant multiple rule: d/dt (3t) = 3
Therefore, ds/dt = 2t + 3
Next, we need to find the derivative of r with respect to t:
dr/dt = d/dt (4t^3 - 27)
To find the derivative of 4t^3, we use the constant multiple rule: d/dt (4t^3) = 12t^2
To find the derivative of 27, we use the constant rule: d/dt (27) = 0
Therefore, dr/dt = 12t^2
Now, we can find ds/dr by dividing ds/dt by dr/dt:
ds/dr = (ds/dt) / (dr/dt)
Substituting the values of ds/dt and dr/dt:
ds/dr = (2t + 3) / (12t^2)
So, the derivative of s with respect to r, ds/dr, is (2t + 3) / (12t^2).