$A$ particle moves in a straight line so that its velocity at any point is given by ${v^2} = a + bx$,where $a, b \neq 0$ are constants. The acceleration is

If $y=5x^2+6x+6$,$x=2$,and $\Delta x=0.001$,then the value of $dy$ is:

$A$ point moves in a straight line during the time $t = 0$ to $t = 3$ according to the law $s = 15t - 2t^2$. The average velocity is

$OA$ and $OB$ are two roads enclosing an angle of $120^{\circ}$. $X$ and $Y$ start from '$O$' at the same time. $X$ travels along $OA$ with a speed of $4 \text{ km/h}$ and $Y$ travels along $OB$ with a speed of $3 \text{ km/h}$. The rate at which the shortest distance between $X$ and $Y$ is increasing after $1 \text{ h}$ is

The total cost $C(x)$ in Rupees,associated with the production of $x$ units of an item is given by $C(x) = 0.005x^{3} - 0.02x^{2} + 30x + 5000$. Find the marginal cost when $3$ units are produced,where marginal cost is defined as the instantaneous rate of change of total cost with respect to the number of units produced.

Find the rate of change of $\sqrt{x^2 + 16}$ with respect to $\frac{x}{x - 1}$ at $x = 3$.