Differentiate the given function.
f(t)= 4(sqrt(t^3)) + 14/sqrt(t) - sqrt(17)
f'(t)=?
This is another calculate-the-derivative problem. "Differentiate" means the same thing. Do it the same way as I explained for the last one you posted as "physics"
To differentiate the given function f(t), we need to find its derivative f'(t).
To find the derivative of each term separately, we can use the power rule, the product rule, and the chain rule.
Let's differentiate each term step by step:
1. Differentiating the first term: 4(sqrt(t^3))
We have a composite function, where the outer function is a constant multiple of the square root function. To differentiate this, we apply the chain rule.
Let u = t^3, then du/dt = 3t^2 (differentiating with respect to t)
Now, we differentiate the outer function. Let f(u) = sqrt(u), then df/du = 1/(2sqrt(u))
Applying the chain rule, we have df/dt = df/du * du/dt
= (1/(2sqrt(u))) * (3t^2)
= (3t^2)/(2sqrt(t^3))
= 3t^2 / (2 * t^(3/2))
= 3/(2 * t^(1/2))
= 3/(2*sqrt(t))
2. Differentiating the second term: 14/sqrt(t)
Using the quotient rule, we can differentiate this term.
Let u = 14 and v = sqrt(t), then du/dt = 0 and dv/dt = 1/(2sqrt(t))
Applying the quotient rule, (u'/v') = (v * du/dt - u * dv/dt) / (v^2)
= (sqrt(t) * 0 - 14 * (1/(2sqrt(t))) ) / (sqrt(t))^2
= -14/(2t)
= -7/t
3. Differentiating the third term: sqrt(17)
The derivative of a constant is always 0.
Now, let's combine all the derivatives to find f'(t):
f'(t) = 3/(2*sqrt(t)) - 7/t + 0
= 3/(2*sqrt(t)) - 7/t
So, the derivative of f(t) = 4(sqrt(t^3)) + 14/sqrt(t) - sqrt(17) is f'(t) = 3/(2*sqrt(t)) - 7/t.