Find the derivative of f(x) = cos x at x = 0
Derivative of a function f(x) at any real number a is given by –
{where h is a very small positive number}
∴ derivative of cos x at x = 0 is given as –
∵ we can’t find the limit by direct substitution as it gives 0/0 (indeterminate form)
So we need to do few simplifications to evaluate the limit.
As we know that 1 – cos x = 2 sin2(x/2)
Dividing the numerator and denominator by 2 to get the form (sin x)/x to apply sandwich theorem, also multiplying h in numerator and denominator to get the required form.
Using algebra of limits we have –
Use the formula:
∴ f’(0) = – 1×0 = 0
Hence,
Derivative of f(x) = cos x at x = 0 is 0