Differentiate the following from first principles
sin x + cos x
We need to find derivative of f(x) = sin x + cos x
Derivative of a function f(x) is given by –
f’(x) = {where h is a very small positive number}
∴ derivative of f(x) = sin x + cos x is given as –
f’(x) =
⇒ f’(x) =
Using algebra of limits we have –
⇒ f’(x) =
We can’t evaluate the limits at this stage only as on putting value it will take 0/0 form. So, we need to do little modifications.
Use: sin A – sin B = 2 cos ((A + B)/2) sin ((A – B)/2) and
cos A – cos B = – 2 sin ((A + B)/2) sin ((A – B)/2)
∴ f’(x) =
Dividing numerator and denominator by 2 in both the terms –
⇒ f’(x) =
Using algebra of limits –
⇒ f’(x) =
Use the formula –
∴ f’(x) =
Put the value of h to evaluate the limit –
∴ f’(x) = cos (x + 0) – sin (x + 0) = cos x – sin x
Hence,
Derivative of f(x) = sin x + cos x = cos x – sin x