Applications of Derivatives revision notes

The derivative $\frac{ds}{dt}$ represents the rate of change of distance $s$ with respect to time $t$. In general, whenever a quantity $y$ varies with another quantity $x$ via a rule $y = f(x)$, the derivative $\frac{dy}{dx}$ (or $f'(x)$) represents the instantaneous rate of change of $y$ with respect to $x$.

• Chain Rule in Rates: If two variables $x$ and $y$ are varying with respect to another variable $t$ (e.g., time), then $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$, provided $\frac{dx}{dt} \neq 0$.
• Sign Convention for Rates: $\frac{dy}{dx}$ is positive if $y$ increases as $x$ increases, and it is negative if $y$ decreases as $x$ increases. JEE Tip Always substitute decreasing rates with a negative sign in equations.
• Marginal Cost (MC): The instantaneous rate of change of total cost $C(x)$ at any level of output $x$. $MC = \frac{dC}{dx}$.
• Marginal Revenue (MR): The rate of change of total revenue $R(x)$ with respect to the number of items sold $x$ at an instant. $MR = \frac{dR}{dx}$.

• Slope of Tangent & Normal: For a curve $y = f(x)$, the slope of the tangent at point $P(x_1, y_1)$ is $m = f'(x_1)$. The slope of the normal is $-\frac{1}{m} = -\frac{1}{f'(x_1)}$.

• Angle of Intersection of Two Curves: The angle $\theta$ between their tangents at the point of intersection. $\tan \theta = |\frac{m_1 - m_2}{1 + m_1 m_2}|$. If $m_1 m_2 = -1$, curves intersect orthogonally.

Let $I$ be an interval in the domain of a real-valued function $f$.

• Increasing: $x_1 < x_2 \implies f(x_1) \le f(x_2)$ for all $x_1, x_2 \in I$.
• Strictly Increasing: $x_1 < x_2 \implies f(x_1) < f(x_2)$ for all $x_1, x_2 \in I$.
• Decreasing: $x_1 < x_2 \implies f(x_1) \ge f(x_2)$ for all $x_1, x_2 \in I$.
• Strictly Decreasing: $x_1 < x_2 \implies f(x_1) > f(x_2)$ for all $x_1, x_2 \in I$.
• Constant: $f(x) = c$ for all $x \in I$.

• Generalized Theorem: If $f'(x) \ge 0$ for $x$ in an interval (and $f'(x)=0$ only at discrete points, not over a sub-interval), the function is strictly increasing. JEE Tip Many students lose marks by incorrectly eliminating endpoints where $f'(x)=0$; discrete zero-derivative points do NOT destroy strict monotonicity (e.g., $f(x)=x^3$ at $x=0$).

• Absolute Maximum/Minimum: A function $f$ has an absolute maximum at $c$ in interval $I$ if $f(c) \ge f(x)$ for all $x \in I$. Absolute minimum if $f(c) \le f(x)$.
• Local Maxima/Minima: A point $c$ is a local maximum if there exists an $h > 0$ such that $f(c) \ge f(x)$ for all $x \in (c-h, c+h)$. Local minimum if $f(c) \le f(x)$ in that neighborhood.
• Critical Point: A point $c$ in the domain of $f$ where either $f'(c) = 0$ or $f'(c)$ is undefined/not differentiable. JEE Tip Sharp corners (like in $f(x) = |x|$) are critical points because the derivative is undefined, and extremum can occur here.

• Differentials: If $y = f(x)$, the differential of $x$ is denoted by $dx = \Delta x$. The differential of $y$ is $dy = f'(x) dx$.
• Approximation Formula: $f(x + \Delta x) \approx f(x) + f'(x) \Delta x$.
• Errors: Absolute error in $x$ is $\Delta x$. Relative error is $\frac{\Delta x}{x}$. Percentage error is $\frac{\Delta x}{x} \times 100$.

• Rolle's Theorem: If $f(x)$ is continuous in $[a,b]$, differentiable in $(a,b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a,b)$ such that $f'(c) = 0$.
• Lagrange's Mean Value Theorem (LMVT): If $f(x)$ is continuous in $[a,b]$ and differentiable in $(a,b)$, then there exists at least one $c \in (a,b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

• Rate of Change: $v = \frac{ds}{dt}$ (Units: ms$^{-1}$)
• Chain Rule for Parametric: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$
• Marginal Cost: $MC = \frac{dC}{dx}$ (Units: Currency/unit)
• Marginal Revenue: $MR = \frac{dR}{dx}$ (Units: Currency/unit)
• Slope of Tangent: $m = \frac{dy}{dx}$
• Equation of Normal: $(y - y_1) = \frac{-1}{(dy/dx)_{x_1, y_1}} (x - x_1)$
• Shortest Distance: Lies along the common normal.
• Extreme Value Theorem applied to Interval $[a, b]$: Calculate $f(c)$ for all critical points $c \in (a, b)$, calculate $f(a)$, and calculate $f(b)$. The absolute max is the largest of these values; absolute min is the smallest.

• Second Derivative Test Failure: When $f'(c) = 0$ AND $f''(c) = 0$, the Second Derivative Test cannot determine the nature of the critical point. You MUST use the First Derivative Test or the $n^{th}$ derivative test,.
• Differentiability Requirement for Tests: Standard tests (first and second derivative) require the function to be differentiable at the point of interest. If a function is not differentiable at $x=c$ (like sharp corners, e.g., $f(x)=|x|$), these derivative tests fail immediately. You must use basic definitions or graphing to find extrema,.
• Domain Restriction: Extreme Value Theorem only guarantees global extrema for continuous functions on closed intervals. In open intervals like $(0, 1)$, a continuous function might have neither a maximum nor a minimum.

• Sign Convention for Monotonicity: $\frac{dy}{dx}$ must be taken as negative if a dimension or quantity is decreasing. Forgetting this negative sign is the #1 error in related rates problems.
• Critical Points vs Local Extrema: ALL local extrema occur at critical points, but NOT ALL critical points are local extrema (e.g., $f(x)=x^3$ at $x=0$).
• Zero Derivative vs Constant: If $f'(x) = 0$ at a specific point, it doesn't mean the function is constant; it just means the tangent is horizontal. For a function to be constant, $f'(x)=0$ must hold for an entire interval.
• Endpoints in Extrema: Endpoints of a closed domain $[a, b]$ can be absolute extrema but are generally NOT considered local extrema because a full open neighborhood $(c-h, c+h)$ does not exist around them.
• Derivative Undefined = Extrema Possible: Many students assume $f'(x) = 0$ is the only condition for extrema. Extrema frequently occur where $f'(x)$ is undefined.

• $y = x^2$ (Parabola): Opens upwards, $f'(x) = 2x$. $x=0$ is the global and local minimum. Decreasing in $(-\infty, 0)$ and increasing in $(0, \infty)$,.
• $y = |x|$ (Modulus Function): V-shaped graph. Minimum at $x=0$. Not differentiable at $x=0$. Demonstrates that an extremum can exist without $f'(0) = 0$.
• $y = x^3$ (Cubic): Passes through origin. $f'(0) = 0$ but $x=0$ is neither a maxima nor a minima; it's a point of inflection. It strictly increases for all real $R$,.
• Turning Points (Hills and Valleys): At local maxima (hills), the function transitions from increasing to decreasing. At local minima (valleys), it transitions from decreasing to increasing.

Algorithm for finding absolute maximum and minimum in $[a, b]$

1. Step 1: Find all critical points in the open interval $(a, b)$ by setting $f'(x) = 0$ and identifying points where $f'(x)$ does not exist.
2. Step 2: Identify the endpoints $x = a$ and $x = b$.
3. Step 3: Calculate the function values $f(x)$ at all points identified in Steps 1 and 2.
4. Step 4: The greatest value computed is the absolute maximum; the least value is the absolute minimum.

Algorithm for Optimization (e.g., maximizing volume)

1. Identify the primary quantity to be maximized/minimized (e.g., Volume $V$),.
2. Express this quantity as a function of a single variable $x$, using given constraints (e.g., $V(x)$),.
3. Find $V'(x)$ and set it to $0$ to find critical points,.
4. Verify using the Second Derivative Test: evaluate $V''(x)$ at the critical point. If $< 0$, it's a maximum; if $> 0$, it's a minimum.

• Geometrical problems with optimization: Finding the maximum area of a rectangle inscribed in an ellipse/circle, or volume of maximum cone in a sphere (Result: Volume is $\frac{8}{27}$ of sphere volume).
• Shortest Distance: Finding minimum distance between a point and a curve (e.g., parabola) or between two curves. Typically requires equating the slopes of normals or maximizing/minimizing the distance formula,.
• Roots of equations: Using Rolle's theorem to prove the existence of roots or find the maximum number of real roots for a given polynomial.
• Strict monotonicity in Trigonometry: Identifying intervals of increase/decrease for composite trigonometric functions (e.g., $\sin x + \cos x$, $\tan^{-1}(\sin x + \cos x)$) by analyzing the sign of trigonometric factors in specific quadrants,.
• Inflection Points & Concavity: Checking conditions for graphs shifting from upwards to downwards opening (important for matching-list questions).