JEE Main Concepts for Application of Derivatives

1. Shortest Distance Between Two Curves

For two smooth non-intersecting curves, the shortest distance generally lies along a common normal. In particular, if the shortest distance is to be found between a curve and a line, then at the nearest point on the curve, the tangent to the curve is parallel to the given line, so that the segment joining the point to the line is perpendicular to both.

2. Monotonicity Involving Unknown Parameters

If a polynomial is required to be increasing on all of R\mathbb{R}, then its derivative must be non-negative for all real values of xx. For a cubic f(x)=ax3+bx2+cx+d,f(x)=ax^3+bx^2+cx+d, we have f′(x)=3ax2+2bx+c.f'(x)=3ax^2+2bx+c. For this quadratic to satisfy f′(x)≥0f'(x)\ge 0 for all real xx, we need:

  • leading coefficient 3a>03a>0, and
  • discriminant ≤0\le 0.

3. Equation of Tangent and Normal in Geometric Problems

The slope of the tangent to y=f(x)y=f(x) at x=x1x=x_1 is f′(x1)f'(x_1). The slope of the normal is the negative reciprocal, −1f′(x1)-\frac{1}{f'(x_1)} whenever f′(x1)≠0f'(x_1)\ne 0. JEE often uses this idea indirectly in shortest-distance and optimization problems.

4. Number of Real Roots Using Monotonicity

If a continuous function is strictly increasing or strictly decreasing on an interval, then it can cross any horizontal line at most once in that interval. This idea is very useful in deciding the number of real roots of equations.

5. Absolute Extrema in Trigonometric Functions

Whenever possible, reduce the trigonometric expression to a single variable form such as a polynomial in sin⁡x\sin x or cos⁡x\cos x, and then use the correct domain of that variable. For example, if t=sin⁡xt=\sin x, then t∈[−1,1]t\in[-1,1].