The below example explains the method of derivative of implicit functions.
Example 1: Find dy / dx, if y + sin y = cos x
dy / dx + cosy . dy / dx = -sin x
dy / dx (1 + cos y) = -sin x
n = 0, 1, 2...
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, sin2y + cos2y = 1
The below example explains the method of derivative of Logarithmic functions
Let y =
The below example explains the method of derivative functions in parametric form.
Example 4: Find, dy / dx, if x = a(θ + sinθ), y = a(1 - cosθ)
dx / dθ = a(1 + cosθ)
dy / dθ = a sin θ
Example 5: Find the value of dy / dx if x2 / 3 + y2 / 3 = a2 / 3
Let x = a cos3θ and y = a sin3θ which satisfy the above equation
dx / dθ = -3a cos2θ sin θ
dy / dθ = 3a sin2θ cosθ
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 d2y / dx2
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]
Note that f’(x) > 0 if x ∈ [0, π / 4) ∪ (5π / 4, 2π] that means f is strictly increasing in this interval