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 may attain a maximum value at a point if its value there is greater than nearby values, and it may attain a minimum value if its value there is smaller than nearby values. These are often called extreme values.

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 absolute greatest or least value a function achieves over an entire closed interval [a, b]. They are also known as the global maximum or global minimum.

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

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

• Local Maxima: f''(c) < 0

• Local Minima: 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 Maxima: f'(x) changes sign from positive to negative as x increases through c.

• Local Minima: f'(x) changes sign from negative to positive as x increases through c.

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

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

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).

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.

• 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.