NCERT Solutions for Class 12 Mathematics Chapter 6 Application of Derivatives

Short Note:

• Derivatives measure the instantaneous rate of change of one quantity with respect to another, useful in problems involving a changing radius, volume, or other physical quantities over time.

• A function is increasing on an interval where f'(x) ≥ 0 throughout, and decreasing on an interval where f'(x) ≤ 0 throughout.

Short Note:

• The slope of the tangent to a curve y = f(x) at a point (x1, y1) is given by f'(x1), and the tangent line's equation is y - y1 = f'(x1)(x - x1).

• The normal to the curve at the same point is perpendicular to the tangent, with slope -1/f'(x1) (when f'(x1) ≠ 0).

Short Note:

• Critical points, where f'(x) = 0 or f'(x) is undefined, are candidates for local maxima or minima and are analyzed using the first or second derivative test.

• Optimization problems (maximizing area, minimizing cost, etc.) are solved by expressing the quantity as a function of a single variable and applying derivative tests to find the optimal value.