What is the derivative
(tˆ2)ˆ(1/3)+2(tˆ5)ˆ(1/2)
what did you get? These are simple power rule problems.
i got [2tˆ(1/3)]+[(10tˆ4)ˆ4
no, you gotta do a little work here.
(t^2)^(1/3) = t^(2/3)
(t^5)^(1/2) = t^5/2
so, you have
y = t^(2/3) + 2t^(5/2)
y' = 2/3 t^(-1/3) + 2(5/2) t^(3/2)
= 2/3 t^(-1/3) + 5t^(3/2)
you need to review your chain rule some more.
Just working it as-is, using the chain rule, you get
if y = u^n, y' = n u^(-1) u'
1/3 (t^2)^(-2/3) * (2t) + 2 (1/2)(t^5)^(-1/2) * (5t^4)
= 2/3 t t^(-4/3) + 5t^4 t^(-5/2)
= 2/3 t^(-1/3) + 5t^(3/2)
To find the derivative of the given expression, which is (t^2)^(1/3) + 2(t^5)^(1/2), we can use the power rule of differentiation.
The power rule states that if we have a function of the form f(x) = x^n, where n is a constant, then the derivative of f(x) is given by f'(x) = n*x^(n-1).
Applying this rule, let's find the derivative of each term separately:
1. (t^2)^(1/3):
Using the power rule, we differentiate t^2 with respect to t: [d/dt (t^2) = 2t]. Then, we multiply the derivative by the exponent of the power: (1/3 * (t^2)^(1/3 - 1)). Simplifying, we have (2t/3) * (t^2)^(-2/3).
2. 2(t^5)^(1/2):
First, we differentiate t^5 with respect to t: [d/dt (t^5) = 5t^4]. Then, multiplying by the exponent of the power, we get (1/2 * 5t^4 * (t^5)^(1/2 - 1)). Simplifying, we have (5t^4) * (t^5)^(-1/2).
Finally, to find the derivative of the given expression, we add our results from step 1 and step 2:
(2t/3) * (t^2)^(-2/3) + (5t^4) * (t^5)^(-1/2).
That is the derivative of the given expression.