Find the velocity function given s(t) = sqrt(t) / (t^2 + 1).
Understand the Problem
The question provides the position function s(t) and asks to find the velocity function. Velocity is the derivative of position (s'(t)). We will need to differentiate the given function s(t) = sqrt(t) / (t^2 + 1) with respect to t.
Answer
$s'(t) = \frac{1 - 3t^2}{2\sqrt{t}(t^2 + 1)^2}$
Answer for screen readers
$s'(t) = \frac{1 - 3t^2}{2\sqrt{t}(t^2 + 1)^2}$
Steps to Solve
- Rewrite the function using exponents
Rewrite the square root as a fractional exponent to make differentiation easier:
$s(t) = \frac{t^{1/2}}{t^2 + 1}$
- Apply the quotient rule
The quotient rule states that if $s(t) = \frac{u(t)}{v(t)}$, then $s'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2}$. In our case, $u(t) = t^{1/2}$ and $v(t) = t^2 + 1$. Thus, we need to find $u'(t)$ and $v'(t)$:
$u'(t) = \frac{1}{2}t^{-1/2}$ $v'(t) = 2t$
Now, we apply the quotient rule:
$s'(t) = \frac{(\frac{1}{2}t^{-1/2})(t^2 + 1) - (t^{1/2})(2t)}{(t^2 + 1)^2}$
- Simplify the expression
Simplify the numerator by multiplying and combining terms:
$s'(t) = \frac{\frac{1}{2}t^{-1/2}(t^2 + 1) - 2t^{3/2}}{(t^2 + 1)^2}$
$s'(t) = \frac{\frac{1}{2}t^{3/2} + \frac{1}{2}t^{-1/2} - 2t^{3/2}}{(t^2 + 1)^2}$
$s'(t) = \frac{-\frac{3}{2}t^{3/2} + \frac{1}{2}t^{-1/2}}{(t^2 + 1)^2}$
- Factor out $\frac{1}{2}t^{-1/2}$ from the numerator
$s'(t) = \frac{\frac{1}{2}t^{-1/2}(-3t^2 + 1)}{(t^2 + 1)^2}$
- Rewrite in terms of square roots and simplify
$s'(t) = \frac{1 - 3t^2}{2\sqrt{t}(t^2 + 1)^2}$
$s'(t) = \frac{1 - 3t^2}{2\sqrt{t}(t^2 + 1)^2}$
More Information
The derivative $s'(t)$ represents the instantaneous rate of change of the position function $s(t)$, which physically means it gives the velocity of an object whose position is described by $s(t)$.
Tips
A common mistake is making an error when applying the quotient rule or forgetting to apply it correctly. Another common mistake is mishandling the exponents during simplification, especially when dealing with fractional and negative exponents. Careful attention to detail is required in each step.
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