fin derivative of the function
(t+e^t)(4- sqrt t)
Use the product rule
d/dt (u*v) = u*dv/dt + v*du/dt
and you get
(t + e^t)[-1/(2*sqrt t)]
+ (1 + e^t)(4 - sqrt t)
Or you could multiply out u(t)*v(t) and differentiate it term by term
u*v = 4t + 4e^t -t^3/2 -e^t*t^1/2
d/dt(u*v) = 4 + 4e^t -(3/2)t^1/2 -e^t*t^1/2 -(1/2)e^t/t^1/2
To find the derivative of the function f(t) = (t + e^t)(4 - sqrt(t)), we can first expand the function using the product rule and then apply the chain rule if necessary.
Let's start by using the product rule: If h(t) = u(t)v(t), then h'(t) = u'(t)v(t) + u(t)v'(t).
In our case, u(t) = (t + e^t) and v(t) = (4 - sqrt(t)). Let's find the derivatives of u(t) and v(t):
1. Derivative of u(t):
u'(t) = derivative of (t + e^t)
= derivative of t + derivative of e^t
= 1 + e^t
2. Derivative of v(t):
v'(t) = derivative of (4 - sqrt(t))
= 0 - 1/(2 * sqrt(t)) [using power rule for differentiation]
= -1/(2 * sqrt(t))
Now, let's apply the product rule:
f'(t) = u'(t)v(t) + u(t)v'(t)
= (1 + e^t)(4 - sqrt(t)) + (t + e^t)(-1/(2 * sqrt(t)))
Simplifying further:
f'(t) = 4(1 + e^t) - (t + e^t)/(2 * sqrt(t)) - e^t(1 + e^t) + (t + e^t)(-1/(2 * sqrt(t)))
Combining like terms:
f'(t) = 4 + 4e^t - (t + e^t)/(2 * sqrt(t)) - e^t - e^(2t) - t/(2 * sqrt(t)) + e^t/(2 * sqrt(t))
Simplifying the expression, we get the derivative of the function:
f'(t) = 4 + 3e^t - t/(2 * sqrt(t)) - e^(2t) + e^t/(2 * sqrt(t))
And that's the derivative of the function (t + e^t)(4 - sqrt(t)).