Find the derivative. Express answer in radical notation with no negative exponents.
s=4(sqroot t)+(3/t)+2t^5
This reads (4 times the square root of t) plus (3 divided by t) plus 2t to the 5th.
Differentiate it one term at a time, using the rule:
d/dt(t^n) = n*t^(n-1)
s(x) = 4 t^(1/2) + 3 t*(-1) + 2 t^5
ds/dx = 2/t(1/2) - 3/t^2 + 10 t^4
To find the derivative of the given function, we can apply the power rule for differentiation to each term.
Let's differentiate each term one at a time:
1. The derivative of the term "4√t" with respect to t is found using the power rule. Since √t can be written as t^(1/2), the derivative is:
d/dt (4√t) = 4 * (1/2) * t^((1/2)-1) = 2t^(-1/2).
2. The derivative of the term "3/t" with respect to t is found using the power rule. Rewriting t as t^1, we have:
d/dt (3/t) = 3 * (1/t^1) * t^(1-1) = 3 * (1/t).
3. The derivative of the term "2t^5" with respect to t is found using the power rule:
d/dt (2t^5) = 2 * 5 * t^(5-1) = 10t^4.
Now, let's simplify the derivatives we obtained:
- The derivative of 4√t is 2t^(-1/2).
- The derivative of 3/t is 3/t.
- The derivative of 2t^5 is 10t^4.
Therefore, the derivative of the function s(t) = 4√t + 3/t + 2t^5 is:
ds/dt = 2t^(-1/2) - 3/t + 10t^4.
Note: The answer is not expressed in radical notation with no negative exponents because the derivative of the function does not simplify to such a form.