APPLICATION OF DERIVATIVES in One Shot: All Concepts & PYQs Covered | JEE Main & Advanced

Summary

The lecture develops applications of derivatives through basic explanations and worked, exam-style problems. Its central theme is to translate a situation into a function or relationship, differentiate appropriately, and interpret the result using signs, critical points, or boundary values.

1. Derivative as a Rate of Change

A derivative with respect to time represents how quickly a quantity changes. The instructor connects this idea to familiar physics quantities, such as velocity as the rate of change of displacement and acceleration as the rate of change of velocity.

General method for related-rate problems:

• Identify the quantities that vary and the rate that is given.
• Write a relationship between the quantities—for example, a geometric formula or a fixed-length constraint.
• Differentiate the relationship with respect to time.
• Substitute the known rate and the values at the specified instant.
• Interpret the sign and units: a negative rate means the quantity is decreasing.

Examples include expanding circular ripples, a ladder sliding down a wall, a leaking balloon, melting ice cream around a spherical object, and water filling a conical vessel.

2. Monotonicity: Increasing and Decreasing Functions

A function is increasing or decreasing according to how its values behave as the input moves through an interval. For differentiable functions, the sign of the derivative provides a practical test:

• Positive derivative: the function is increasing.
• Negative derivative: the function is decreasing.
• Zero derivative: check the sign on either side; zero alone does not determine the behavior.

For interval questions, the instructor recommends:

• Find and factor the derivative.
• Identify critical values and points excluded from the domain.
• Test the derivative’s sign on each resulting interval, often using a sign chart or “wavy curve” method.
• Report increasing and decreasing intervals without including points outside the domain.

The lecture also stresses that increasing and decreasing behavior can be defined directly from function values, so the derivative test is only a tool for differentiable functions.

Additional approaches include:

• If equations involving both (f(x)) and (f(1/x)) are given, replace (x) by (1/x) to obtain a second equation, then solve the pair.
• If (f’) is known to be increasing, inequalities between values of (f’) can be converted into inequalities between their inputs.
• For a parameter that must make a function monotonic for every (x), impose the appropriate derivative inequality for all (x), then use the maximum or minimum of the remaining expression to constrain the parameter.
• To compare expressions such as (e^\pi) and (\pi^e), determine where an auxiliary function increases or decreases, then compare the relevant inputs.

3. Critical Points

A critical point occurs at a point in the function’s domain where the derivative is zero or does not exist. Sharp corners can therefore be critical points even when the function itself is continuous.

A point where the derivative is zero is not automatically a maximum or minimum; the surrounding behavior must still be checked.

4. Local and Global Maxima and Minima

A local maximum or minimum is the greatest or least function value in a neighborhood of a point. A global (absolute) maximum or minimum is the greatest or least value across the entire domain.

The lecture emphasizes these distinctions:

• A global maximum is also a local maximum, and a global minimum is also a local minimum.
• The reverse is not necessarily true.
• Local extrema are determined by nearby behavior, not by comparing their values with extrema elsewhere.
• At a boundary point, the neighborhood may exist on only one side, so a boundary can still be a local extremum.

First-Derivative Test

Check how (f’) changes sign as (x) passes through a critical point:

• Negative to positive: local minimum.
• Positive to negative: local maximum.
• No sign change: neither a local maximum nor a local minimum.

The instructor presents this as a dependable method, including at continuous sharp corners where the derivative at the point itself may not exist.

Second-Derivative Test

At a differentiable critical point where (f’(a)=0):

• (f’‘(a)>0): local minimum.
• (f’‘(a)<0): local maximum.
• (f’‘(a)=0): the test is inconclusive; return to the first-derivative test or another method.

The lecture explains this through the way the first derivative changes around a maximum or minimum.

Finding Global Extrema on a Closed Interval

For a continuous function on a closed interval:

• Find critical points within the interval.
• Evaluate the function at those points and at both endpoints.
• The largest and smallest of the resulting values are the global maximum and minimum.

This also provides a method for finding the range on such an interval.

5. Optimization Problems

Many optimization questions begin with several variables. The key is to use the given constraint to express the objective as a function of one variable, then differentiate and check which candidate gives the required maximum or minimum.

Examples involve choosing dimensions to maximize a product, minimizing or maximizing a box’s dimensions, and optimizing volumes or surface areas of shapes such as cylinders and spheres.

General optimization workflow:

• Define variables for the unknown dimensions or quantities.
• Write the constraint relating them.
• Express the quantity to be optimized in terms of one variable.
• Differentiate and solve for stationary points.
• Check feasibility, endpoints, and whether the candidate is a maximum or minimum.
• Substitute the result back to find the requested dimensions or value.

The instructor also shows:

• Using AM–GM as an alternative when the quantities are positive and the expression fits the inequality.
• Differentiating a constraint and an objective with respect to one variable when eliminating variables directly would be cumbersome.

6. Concavity and Points of Inflection

Concavity describes the direction in which a graph bends:

• (f’‘(x)>0): concave upward.
• (f’‘(x)<0): concave downward.

A point of inflection is where concavity changes. Solving (f’‘(x)=0) gives candidates, but it does not by itself establish an inflection point; the sign of (f’‘) must change across the point. The first derivative at an inflection point may or may not be zero.

The lecture uses examples including powers of (x), cosine, sine, and (\sin^2 x), illustrating that a graph’s shape depends on concavity changes and that (\sin^2 x) is not simply the same graph as (|\sin x|).

7. Sketching Cubic Graphs

To analyze a cubic, differentiate it and study the resulting quadratic:

• If the derivative keeps one sign, the cubic is monotonic and has no local maximum or minimum.
• If the derivative changes sign at critical points, the cubic has local extrema.
• The second derivative can help locate inflection points.
• A derivative-zero point does not necessarily correspond to a distinct (x)-intercept; the cubic may touch the (x)-axis at a repeated root.
• To determine how many times the graph meets the (x)-axis, combine critical-point information with function values at those points and the graph’s overall behavior.

The provided subtitles end during the cubic-graph discussion. Although the lecture’s opening announces further topics, the supplied excerpt does not include the subsequent explanations of tangents, normals, angle between curves, minimum distance, or the mean value theorem.

Speakers and Sources

• Instructor: The sole speaker addressing the class; identified in the video metadata as Tarun Khandelwal Sir.
• Source: JEE Wallah’s Manzil lecture series. No guest speakers are featured.