Revision: Mathematics >> Applications of Derivatives CUET (UG) Applications of Derivatives

If two variables x and y both vary with respect to a third variable t (like time), you can find the rate of change of y with respect to x using:

(Note: This is only valid if \[\frac{dx}{dt} \neq 0\]).

If a quantity y varies with another quantity x based on a rule y = f(x), then the derivative \[\frac{dy}{dx}\] (or f'(x)) represents the rate of change of y with respect to x.

Evaluating the derivative at a specific point, \[\left.\frac{dy}{dx}\right|_{x=x_0}\], gives the instantaneous rate of change at exactly \[x = x_0\].

A function f(x) is said to be an increasing function on (a, b) if x₁ < x₂ ⇒ f(x₁) ≤ f(x₂)

Strictly Increasing Function:

• If x₁ < x₂ ⇒ f(x₁) < f(x₂)

A function f(x) is said to be a decreasing function on (a, b) if x₁ < x₂ ⇒ f(x₁) ≥ f(x₂)

Strictly Decreasing Function:

• If x₁ < x₂ ⇒ f(x₁) > f(x₂)

A function f is said to be monotonic in an interval if it is either increasing or decreasing in that interval.

A function f is said to be constant on I if f(x) = c for every x ∈ I, where c is a constant.

Let x₀ be a point in the domain of a real-valued function f.

The function f is said to be increasing at x₀ if there exists an open interval containing x₀ in which f is increasing.

Similarly, f is said to be decreasing at x₀ if there exists an open interval containing x₀ in which f is decreasing.

A point in the domain of a function is called a critical point if either the derivative is zero there or the derivative does not exist there. Critical points are checked while locating possible maxima or minima.

The points where a function changes from decreasing to increasing or from increasing to decreasing are called turning points.

Let f be a function defined on an interval I.

Maximum value: f has a maximum value at c ∈ I if \[ \boxed{f(c) \geq f(x) \quad \text{for all } x \in I} \]

The value f(c) is called the maximum value and c is called a point of maximum.

Minimum value: f has a minimum value at c ∈ I if \[ \boxed{f(c) \leq f(x) \quad \text{for all } x \in I} \]

The value f(c) is called the minimum value and c is called a point of minimum.

Extreme value: A maximum or minimum value of f is called an extreme value.

 
           Maximum value                          Minimum value

Absolute Maximum: The greatest value of a function on the entire given interval is called its absolute maximum value or global maximum value.

Absolute Minimum: The least value of a function on the entire given interval is called its absolute minimum value or global minimum value.

\[\mathrm{f(a+h)\approx f(a)+h~f^{\prime}(a)}\]

Assume f'(c) = 0 and the second derivative exists at c:

• Local Maximum: f''(c) < 0

• Local Minimum: f''(c) > 0

• Test Fails: f''(c) = 0. If this happens, you must go back and use the First Derivative Test to check if it is a maxima, minima, or point of inflection.

Let c be a critical point of a continuous function f:

• Local Maximum: If f'(x) changes sign from positive to negative as x passes through c, then cc is a point of local maximum.

• Local Minimum: If f'(x) changes sign from negative to positive then c is a point of local minimum.

• Point of Inflection: f'(x) does not change sign as x passes through c (it is neither a maxima nor a minima).

If a differentiable function has an absolute max or min at an interior point c of the interval, then its derivative at that point is zero (f'(c) = 0).

A continuous function on a closed interval [a, b] will attain its absolute maximum and absolute minimum value at least once in that interval.

Given:

y `sqrt(x^2 + 1) = log (sqrt(x^2 + 1) - x)`

Differentiate the Left-Hand Side:

Using the product rule (uv)′ = u′v + uv′:

Let u = y and v = `sqrt(x^2 + 1)`

`d/dx (y sqrt(x^2 + 1)) = (dy)/(dx) . sqrt(x^2 + 1) + y . d/dx (sqrt(x^2 + 1))`

= `sqrt(x^2 + 1) (dy)/(dx)  + y . (1/(2sqrt(x^2 + 1)) . 2x)`

= `sqrt(x^2 + 1) (dy)/(dx) + (xy)/sqrt(x^2 + 1)`   ...(i)

Differentiate the Right-Hand Side:

Using the chain rule for log(u):

`d/dx [log (sqrt(x^2 + 1) - x)] = 1/(sqrt(x^2 + 1) - x) . d/dx (sqrt(x^2 + 1) - x)`

= `1/(sqrt(x^2 + 1) - x) . (x/sqrt(x^2 + 1) - 1)`

Take the LCM in the bracket:

= `1/(sqrt(x^2 + 1) - x) . ((x - sqrt(x^2 + 1))/sqrt(x^2 + 1))`

= `1/(sqrt(x^2 + 1) - x) . ((-sqrt(x^2 + 1) - x)/sqrt(x^2 + 1))`

= `-1/(sqrt(x^2 + 1)`   ...(ii)

Equate LHS and RHS

`sqrt(x^2 + 1) (dy)/(dx) + (xy)/sqrt(x^2 + 1) = -1/(sqrt(x^2 + 1)`

Multiply the entire equation by `sqrt(x^2 + 1)` to clear the denominators:

`(sqrt(x^2 + 1) . sqrt(x^2 + 1)) (dy)/(dx) + xy = -1`

`(x^2 + 1) (dy)/(dx) + xy = -1`

`(x^2 + 1) (dy)/(dx) + xy + 1 = 0`

Hence proved

• Derivative gives instantaneous rate of change.

• Positive derivative means the quantity is increasing.

• Negative derivative means the quantity is decreasing.

• In related rates, first connect the variables by an equation, then differentiate.

• Always substitute the given value only after differentiation.

• Do not forget units in the final answer.

• Marginal cost and marginal revenue are applications of derivatives in economics.

• Increasing means output does not decrease as input increases.
• Strictly increasing means output always increases.
• Decreasing means output does not increase as input increases.
• Monotonic means either increasing or decreasing on an interval.
• f′(x) > 0 implies increasing, f′(x) < 0 implies decreasing, and f′(x) = 0 on an interval implies constant behaviour.
• If \[ f'(x) = 0 \] throughout an interval, the function is constant on that interval.
• A single point where \[ f'(x) = 0 \] does not necessarily make the function constant.
• To find intervals of increase or decrease, find the zeros of f'(x), divide the domain into intervals, and check the sign of f'(x).
• A function that is increasing or decreasing on an interval is called monotonic on that interval.
• A function may be increasing on one interval and decreasing on another; in that case it is not monotonic on its entire domain.

• Maxima and minima are extreme values of a function.

• Critical points occur where \(f'(x)=0\) or \(f'(x)\) is not defined.

• If \(f'(x)\) changes from positive to negative, the function has a local maximum.

• If \(f'(x)\) changes from negative to positive, the function has a local minimum.

• If \(f''(c) < 0\), there is a local maximum at \(x=c\).

• If \(f''(c) > 0\), there is a local minimum at \(x=c\).

• For absolute extrema on \([a,b]\), compare values at critical points and endpoints.

• Not every critical point gives a maximum or minimum.

• The second derivative test is quick, but the first derivative test is often more reliable in detailed reasoning.

• Continuity on a closed interval guarantees existence of absolute extrema.

• Differentiability at an interior extremum implies \(f'(c)=0\).

• Endpoints must always be checked in closed interval problems.

• Local extrema and absolute extrema are not always the same.

• A critical point occurs when \(f'(x)=0\) or \(f'(x)\) is undefined.

• Optimisation problems in calculus are applications of maxima and minima.