$A$ triangle has two fixed vertices $A(a, 0)$ and $B(0, b)$. Let its third vertex $C$ move along the line $x = y$. If $s$ is the area of triangle $ABC$, then $\frac{ds}{dx} =$

If a balloon lying at an altitude of $30 \ m$ from an observer at a particular instant is moving horizontally at the rate of $1 \ m/s$ away from him, then the rate at which the balloon is moving away directly from the observer at the $40^{\text{th}}$ second is (in $m/s$)

The rate of change of the diagonal $R$ of a square with respect to its area $A$ 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.

If the velocity of a moving particle is directly proportional to the square root of the distance it has covered,what is its acceleration?

The displacement $x$ of a particle at time $t$ is given by $x = At^2 + Bt + C$, where $A, B, C$ are constants and $v$ is the velocity of the particle. Then the value of $4Ax - v^2$ is: