The document Differentiation - Derivatives Notes | Study Engineering Mathematics - GATE is a part of the GATE Course Engineering Mathematics.

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**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****Solution:**

dy / dx + cosy . dy / dx = -sin x

dy / dx (1 + cos y) = -sin x** **n = 0, 1, 2...

**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

We know, sin^{2}y + cos^{2}y = 1

Note:

**Logarithmic Differentiation**

The below example explains the method of derivative of Logarithmic functions

**Example 3:****Solution:**

Let y =

**Derivatives of Functions in Parametric Forms **

The below example explains the method of derivative functions in parametric form.

**Example 4: Find, dy / dx, if x = a(θ + sin θ), y = a(1 - cosθ)**

dx / dθ = a(1 + cosθ)

dy / dθ = a sin θ

**Example 5: Find the value of dy / dx if x ^{2 / 3} + y^{2 / 3} = a^{2 / 3} **

Let x = a cos

dx / dθ = -3a cos

dy / dθ = 3a sin

**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.

Let y = f (x) then dy / dx = f' (x) ...........(1)

If f’(x) is differentiable, we may differentiate above equation w.r.t x.

i.e. d / dx (dy / dx) is called the second order derivative of w.r.t x and it is denoted by d^{2}y / dx^{2}

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)

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

Now, f' (x) = 0 gives sin x = cos x which gives that x = π / 4, 5π / 4 in 0 ≤ x ≤ 2π

The point x = π / 4 and x = 5π / 4 divide the interval[0, 2π] into three disjoint intervals

Namely [0, π / 4), (π / 4, 5π / 4) and (5π / 4, 2π]

Notethat f’(x) > 0 if x ∈ [0, π / 4) ∪ (5π / 4, 2π] that means f is strictly increasing in this interval

The document Differentiation - Derivatives Notes | Study Engineering Mathematics - GATE is a part of the GATE Course Engineering Mathematics.

All you need of GATE at this link: GATE