DerivativesJEE Main

Concept

The derivative $f'(x_0)$ is the slope of the tangent at $x_0$ — defined as the limit of secant slopes as the gap $h$ shrinks to zero. It measures the instantaneous rate of change.

Key formula

Derivation

For $f(x)=kx^2$: $f(x_0+h)=k(x_0+h)^2=k(x_0^2+2x_0h+h^2)$.

The difference quotient is $\dfrac{f(x_0+h)-f(x_0)}{h}=\dfrac{k(2x_0h+h^2)}{h}=k(2x_0+h)$.

As $h\to 0$ this approaches $2kx_0$. Hence $f'(x_0)=2kx_0$, matching the power rule $\frac{d}{dx}(kx^2)=2kx$.

Scenarios to explore

• Derivatives — Tangent slope as the limit of secants.

Real-world applications

• Velocity = derivative of displacement; acceleration = derivative of velocity.
• Finding maxima/minima where f'(x) = 0.
• Linear approximation: f(x₀+h) ≈ f(x₀) + f'(x₀)·h.

JEE exam tips

• Differentiability ⇒ continuity, but not the converse (e.g. |x| at 0).
• Left-hand and right-hand derivatives must match for f'(x) to exist.

Common mistakes

• Forgetting the limit — a secant slope is only an approximation until h → 0.
• Dropping the chain rule for composite functions.
• Assuming f'(x₀)=0 means a maximum; it can be a minimum or inflection.

Exam traps to avoid

• A sharp corner has no tangent — the derivative does not exist there.
• f'(x₀)=0 is necessary but not sufficient for an extremum.