Differentiation - GATE Study Material in PDF

Differentiation - GATE Study Material inPDF Differentiation is one the two important operations, along with Integration, in Calculus. These Free Study Notes are important for GATE EC, GATE EE, GATE ME, GATE CS, GATE CE as well as other exams like BARC, BSNL, IES, DRDO etc. These GATE Study Material can be downloaded in PDF so that your exam preparation is made easy and you ace your paper. Before you start, make sure you go through the basics of Engineering Mechanics though. Recommended Reading – Types of Matrices Properties of Matrices Rank of a Matrix & Its Properties Solution of a System of Linear Equations Eigen Values & Eigen Vectors Linear Algebra Revision Test 1 Laplace Transforms Limits, Continuity & Differentiability If the value of f'(x) is finite on (a, b) then it is said to be derivable on (a, b). The derivatives of various functions with examples are given below. Derivatives of Implicit Functions The below example explains the method of derivative of implicit functions. Example 1: Find dy dx , if y + sin y = cos x 1|Page Solution: dy dy + cos y ⋅ dx = − sin x dx dy dx (1 + cos y) = − sin x dy − sin x = 1+cos y , where y ≠ (2n + 1) π n = 0, 1, 2 … dx Derivatives of Inverse Trigonometric Functions The below example explains the method of derivative of Inverse Trigonometric functions. Example 2: Find the value of f’(x) where f(x) = sin-1 x Solution: Let y = sin-1 x ⇒ sin y = x dy cos y dx = 1 dy dx 1 = cos y We know, sin2y + cos2y = 1 cos y = √1 − sin2 y cos y = √1 − x 2 dy dx 1 = √1−x2 Note: d dx (tan−1 x) = 2|Page 1 1+x2 Logarithmic Differentiation The below example explains the method of derivative of Logarithmic functions Example 3: Differentiate, √ (x−3)(x2 +4) 3x2 +4x+5 with respect to x. Solution: (x−3)(x2 +4) let y = √ 3x2 +4x+5 1 log y = 2 [log(x − 3) + log(x 2 + 4) − log(3x 2 + 4x + 5)] 1 dy 1 1 2x 6x+4 = 2 [x−3 + x2 +4 − 3x2 +4x+5] y dx dy 1 (x−3)(x2 +4) = 2√ dx 3x2 +4x+5 1 2x 6x+4 [x−3 + x2 +4 − 3x2 +4x+5] Derivatives of Functions in Parametric Forms The below example explains the method of derivative functions in parametric form. Example 4: dy Find, dx , if x = a(θ + sin θ), y = a(1 − cos θ) Solution: dx dθ dy dθ dy dx = a(1 + cos θ) = a sin θ = dy dθ dx dθ a sin θ = a(1+cos θ) 3|Page dy dx dy dx = θ 2 θ 2 2 sin cos 2 cos2 θ 2 θ = tan 2 Example 5: Find the value of dy dx 2 2 2 if x 3 + y 3 = a3 Solution: Let x = a cos 3 θ and y = a sin3 θ which satisfies the above equation dx dθ dy dθ = −3a cos 2 θ sin θ = 3a sin2 θ cos θ dy = dx dy dθ dx dθ 3 y = − tan θ = − √x Second Order Derivative So far we have seen only first order derivatives and second order derivative can be obtained by again differentiating first order differential equation with respect to x. dy Let y = f(x)then dx = f′(x) − − − − − − (1) If f’(x) is differentiable, we may differentiate above equation w.r.t x. i. e. dy d2 y dx dx dx2 d ( ) is called the second order derivative of y w. r. t x and it is denoted by Note: Let the function f(x) be continuous on [a, b] and differentiable on the open interval (a, b), then 1. f(x) is strictly increasing in [a, b] if f’(x) > 0 for each x ∈ (a, b) 4|Page 2. f(x) is strictly decreasing in [a, b] if f’(x) < 0 for each x ∈ (a, b) 3. f(x) is constant function in [a, b] if f’(x)= 0 for each x ∈ (a, b) Example 6: Find the intervals in which the function f is given by f(x)= sin x + cos x : 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing. Solution: We have, f(x) = sin x + cos x f'(x) = cos x – sin x π 5π Now, f′(x) = 0 gives sin x = cos x which gives that x = 4 , The point x = π π 4 and x = π 5π Namely [0, 4 ) , (4 , 4 5π 4 4 in 0 ≤ x ≤ 2π divide the interval [0, 2π] into three disjoint intervals, 5π ) and ( 4 , 2π] π 5π Note that f′(x) > 0 if x ∈ [0, 4 ) ∪ ( 4 , 2π] that means f is strictly increasing in this interval π 5π Also, f′(x) < 0 if x ∈ ( 4 , 4 ) that means f is strictly decreasing in this interval In the next article we will see about Partial Differentials. Did you like this article on Differentiation? Let us know in the comments? You may also like the following articles – Try out Calculus on Official GATE 2017 Virtual Calculator Recommended Books for Engineering Mathematics 40+ PSUs Recruiting through GATE 2017 Partial Differentiation 5|Page