In this article, you will learn what is the derivative of cos x as well as prove the derivative of cos x by quotient rule, first principal rule, and chain rule.
So, without wasting time let's get started.
What is cos x?
Cos x is a trigonometric function that is reciprocal of sec x.
Derivative of cos x
The derivative of cos x is equal to the negative of sin x.
We can prove the derivative of cos x in three ways first by using the quotient rule and second by using the first principle rule and the last chain rule.
Derivative of cos x Proof by Quotient Rule
The formula of the quotient rule is,
dy/dx = {v (du/dx) - u (dv/dx)}/v²
Where,
dy/dx = derivative of y with respect to x
v = variable v
du/dx = derivative of u with respect to x
u = variable u
dv/dx = derivative of v with respect to x
v = variable v
Let us,
y = cos x
As we know,
cos x = 1/sec x
So,
We can written as,
y = 1/sec x.
Where,
u = 1
v = sec x
Now putting these values on the quotient rule formula, we will get
dy/dx = [sec x d/dx(1) - 1 d/dx(sec x)] / (sec x)²
Since,
d/dx (sec x) = sec x .tan x and d/dx (1) = 0
So,
dy/dx = (0 - sec x . tan x )/ (sec x)²,
dy/dx = - tan x/sec x
As we know,
tan x = (sin x/ cos x) and sec x = 1/cos x
So,
dy/dx = - (sin x/ cos x)/(1/cos x)
d/dx ( cos x) = - sin x
Thus, we proved the derivative of cos x will be equal to - sin x using the quotient rule method.
Derivative of cos x Proof by First Principle Rule
According to the first principle rule, the derivative limit of a function can be determined by computing the formula:
For a differentiable function y = f (x)
We define its derivative w.r.t x as :
dy/dx = f ' (x) = limₕ→₀ [f(x+h) - f(x)]/h
f'(x) = limₕ→₀ [f(x+h) - f(x)]/h
This limit is used to represent the instantaneous rate of change of the function f(x).
Let,
f (x) = cos x
So,
f(x + h) = cos (x + h)
Putting these values on the above first principle rules equation.
f' (x) = limₕ→₀ [cos (x + h) - cos x]/h
So, as we know
cos (a + b) = cos a cos b - sin a sin b
f' (x) =limₕ→₀[cos x.cos h - sin x.sin h - cos x]/h
= limₕ→₀[ {(cos h - 1)/h}cos x - (sin h/h)sin x]
= limₕ→₀( 0.cos x - 1 . sin x)
f' (x) = - sin x
Thus, we proved the derivative of cos x will be equal to - sin x using the first principle rule method.
Derivative of cos x Proof by Chain Rule
Let us,
y = cos x
As we know,
sin{(π/2) - x} = cos x
So,
y = sin{(π/2) - x}
By using the chain rule,
The formula of chain rule is,
dy/dx = (dy/du) × (du/dx)
Where,
dy/dx = derivative of y with respect to x
dy/du = derivative of y with respect to u
du/dx = derivative of u with respect to x
After putting these values we can find,
dy/dx = d/dx [sin{(π/2) - x}]
Since, d/dx(sin x) = cos x
dy/dx = [cos{(π/2) - x}] . (-1)
Since, cos{(π/2) - x} = sin x
Hence,
d/dx ( cos x) = - sin x
Thus, we proved the derivative of cos x will be equal to - sin x using the chain rule method.
So friends here I discussed all aspects related to the derivative of cos x.
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