Q.Differentiate $f(x) = x^3$ from first principles.

Increasing at $x=a\Rightarrow f'(a)>0$.

The sign of the first derivative gives the direction of change of a function. If $f(x)$ is increasing at $x=a$, the tangent slopes upward, so $f'(a)>0$. (A negative derivative means decreasing, and a zero derivative means a stationary point.)

Decreasing at $x=a\Rightarrow f'(a)<0$.

The sign of the first derivative indicates whether a function is rising or falling. If $f(x)$ is decreasing at $x=a$, the tangent line slopes downward, so the rate of change is negative:

$f'(a)<0.$

The formula for price elasticity of demand is $\eta = -\dfrac{P}{x}\cdot\dfrac{dx}{dp}$ — option (a).

GSEB Class-12 Statistics, Differentiation (business applications):

Elasticity of demand measures the proportionate change in quantity demanded $x$ relative to the proportionate change in price $P$:

$\eta = -\frac{P}{x} \cdot \frac{dx}{dp}$

The negative sign is included because $\dfrac{dx}{dp}$ is normally negative (demand decreases as price rises), which makes the elasticity a positive number.

Elasticity of demand $\eta = -\dfrac{p}{x}\cdot\dfrac{dx}{dp}$.

Elasticity of demand measures how sensitively the quantity demanded responds to a change in price. If $x$ is the quantity demanded at price $p$, it is defined as the ratio of the proportionate change in demand to the proportionate change in price, which in derivative form is

$\eta = -\frac{p}{x}\cdot\frac{dx}{dp}$

Since demand normally decreases as price increases, $\dfrac{dx}{dp}$ is negative; the minus sign is included so that the elasticity $\eta$ comes out positive.