In this article, we shall learn about the properties of derivatives and their application to find the maxima and minima. After that, we begin with integral calculus – indefinite integrals, properties, and methods of finding the integral of a function.
Also, check: Differentiation of a Function.
Check the latest NEET Syllabus for the 2023/2024 exams.
Table of Contents:
- Properties of Derivatives
- Increasing and Decreasing Functions
- Maxima & Minima
- Indefinite Integration
- Formulas
Properties of Derivatives
Following are the properties of derivatives that are very helpful while determining the derivative of a given function.
Derivative of a Constant
Let a be any real constant number and f be a function x then,
For example, derivative of 5sin x is
d/dx (5sin x) = 5 d/dx(sin x) = 5 cos x.
Derivative of Sum or Difference of a Function
Suppose we have to find the derivative of the sum or difference of two functions of x, f(x) and g(x), then,
For example, let us find the derivative of 4x2 + cosec x
d/dx [4x2 + cosec x] = d/dx(4x2) + d/dx(cosec x) = 8x – cosec x cot x.
Product Rule
The derivative of the product of two functions, f(x) and g(x,) is determined by
For example, let us find the derivative of x3cos x
d/dx(x3 cos x) = cos x d/dx(x3) + x3 d/dx(cos x)
= cos x. 3x2 + x3 (– sin x) = 3x2 cos x – x3 sin x
Quotient Rule
If two functions of x, f(x) and g(x) are in are in a division form, then
For example, the derivative of (sin x)/(x2 + 1) is calculated as
Double Differentiation
Double differentiation is also known as the second-order derivative of a function. Double differentiation means differentiating the derivative of a function again. Let y = f(x) be a real-valued function, then the second order derivative of f is given by
Double differentiation of f is also written as f’’(x).
The graphical interpretation of the second-order derivative of a function is that if the first-order derivative represents the slope of the function at a point, then the second-order derivative represents the rate of change of the slope of the graph.
Let us take an example, let f(x) = sin x, then the second-order derivative of f can be calculated as
In this way, differentiating successively, we can find other higher-order derivatives of a given function.
Increasing and Decreasing Functions
One of the applications of derivatives is to find the nature of a function, whether it is increasing or decreasing.
A function f(x) is said to be an increasing function, if for any x1 < x2 we have f(x1) ≤ f(x2) which means for increasing values of x, the value of f(x) also increases.
In contrast,a function f(x) is said to be a decreasing function, if for any x1 < x2 we have f(x1) ≥ f(x2) which means for increasing values of x, the value of f(x) decreases.
For example, f(x) = ex is an increasing function.
And f(x) = e-x is a decreasing function.
Now we may come across functions that increase for a certain interval and decrease for another. For example, f(x) = sin x,
We observe that for x ∈ [0, 𝜋/2], the function increases, and after that, it decreases. It also has some points where the tangent to the function has zero slope, that is, f’(x) = 0. These points are called critical points.
For increasing parts of the function, f’(x) > 0 and for decreasing parts of the function, f’(x) < 0.
Thus, to sum up:
|
If f’(x) > 0, |
The function has an increasing tangent slope, that is, the function is increasing. |
|---|---|
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If f’(x) = 0, |
The tangent to the function at x is parallel to the x-axis, and x is called a critical point of f. |
|
If f’(x) < 0, |
The function has a decreasing tangent slope, that is, the function is decreasing. |
Maxima and Minima
Maxima and minima are points in the domain of a function where the function attains maximum and minimum values, respectively.
Second Order Derivative Test
We shall use derivatives to determine local maxima and minima for a given function.
For a function f, if x = a is a maxima, then as the value of x increases after a, the slope of the tangent to the graph decreases. If x = a is a minima, then as the value of x increases after a, the slope of the tangent to the graph increases.
Thus, the second order derivative test is given as:
- Let x = a be a critical point of f such that f’(x) = 0 at x = a, then if f’’(x) < 0, x = a is a point of local maxima.
- Let x = a be a critical point of f such that f’(x) = 0 at x = a, then if f’’(x) > 0, x = a is a point of local minima.
- If f’’(a) = 0, then the test fails.
Steps to find maxima and minima of a given function:
- Let f(x) be the given function, find the first order derivative of f and equate it to zero.
- Solve the equation to get the value of x. Let x = a be the critical point of f.
- Determine the second order derivative of f at x = a.
- If f’’(a) < 0, x = a is a point of local maxima.
- If f’’(a) > 0, x = a is a point of local minima.
- If f’’(a) = 0, then the test fails.
For example, a ball is thrown up in the air. At any instant t, its height is given the equation h = 3 + 14t – 5t2. Let us find the maximum height attained by the ball using the concept of maxima and minima.
The maximum height hmax will be the maximum point of the graph of h = 3 + 14t – 5t2.
Now, h’(t) = 14 – 10t
Equating the above equation to zero, we get
h’(t) = 0
⇒ 14 – 10t = 0
⇒ t = 14/10 = 1.4s
Let us check for maxima and minima at t = 1.4 s.
h’’(t) = –10
∴ h’’(1.4) = –10 < 0.
Thus, t = 1.4 s is a point of maxima and the maximum height attained is hmax = 3 + 14 × 1.4 – 5 (1.4)2 = 12.8 m.
Indefinite Integration
Integration is the reverse process of differentiation. If F is the derivative of f with respect to x, then
∫ F(x) dx = f(x) + C where f’(x) = F(x) and C is the constant of integration.
That is, f(x) is the integral of F. Hence, integration is often referred to as anti-derivatives or primitives.
Formulae for Integration
Following are the formulae we often use while finding the integral of a function. We can compare the integration and differentiation of functions side-by-side for a better understanding.
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Derivative |
Integral |
|---|---|
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\(\begin{array}{l}\frac{d}{dx}(c)=0\:where\:c\:is\:a\:constant\end{array} \)
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∫ 0 dx = c |
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\(\begin{array}{l}\frac{d}{dx}(kx)=k\end{array} \)
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∫ k dx = kx + c |
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\(\begin{array}{l}\frac{d}{dx}(sin\;x)=cos\;x\end{array} \)
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∫ cos x dx = sin x + c |
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\(\begin{array}{l}\frac{d}{dx}(cos\;x)=-sin\;x\end{array} \)
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∫ sin x dx = –cos x + c |
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\(\begin{array}{l}\frac{d}{dx}(tan\;x)=sec^{2}\;x\end{array} \)
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∫ sec2 x dx = tan x + c |
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\(\begin{array}{l}\frac{d}{dx}(cot\;x)=-cosec^{2}\;x\end{array} \)
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∫ cosec2 x dx = –cot x + c |
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\(\begin{array}{l}\frac{d}{dx}(sec\;x)=sec\;x\;tan\;x\end{array} \)
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∫ sec x tan x dx = sec x + c |
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\(\begin{array}{l}\frac{d}{dx}(cosec\;x)=-cosec\;x\;cot\;x\end{array} \)
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∫ cosec x cot x dx = –cosec x + c |
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\(\begin{array}{l}\frac{d}{dx}(ln\;x)=\frac{1}{x}\end{array} \)
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∫ (1/x) dx = ln x + c |
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\(\begin{array}{l}\frac{d}{dx}(e^{x})=e^{x}\end{array} \)
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∫ ex dx = ex + c |
Some rules regarding integration of polynomial functions:
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