Find the derivative of cos x from first principle.
Consider the given function,
f ( x ) = cos x
According to the first principle, the derivative of a function is,
f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h
Applying the above formula to the given function,
f ′ ( x ) = lim h → 0 cos ( x + h ) − cos x h f ′ ( x ) = lim h → 0 cos x cosh − sin x sinh − cos x h f ′ ( x ) = lim h → 0 [ − cos x ( 1 − cosh ) − sin x sinh h ] f ′ ( x ) = − cos x lim h → 0 ( 1 − cosh ) h − sin x lim h → 0 sinh h (1)
From the formula of limits, we know that,
lim x → 0 1 − cos x x = 0
And,
lim x → 0 sin x x = 1
Therefore, equation (1) becomes,
f ′ ( x ) = − cos x ( 0 ) − sin x ( 1 ) = 0 − sin x = − sin x
Thus, the derivative of cos x is − sin x .
Consider the given function,
According to the first principle, the derivative of a function is,
Applying the above formula to the given function,
From the formula of limits, we know that,
And,
Therefore, equation (1) becomes,
Thus, the derivative of
