Question

f(x)cos x can you differentiate this by using first principle of differentiation

Answer 1

Answer: - sin (x)

Step-by-step explanation:

The required formula:

i)cos (a+b) = cos(a) cos(b) – sin(a) sin(b)

ii)  $\lim_{x \to\ 0}$ [{cos(x) – 1} / x] = 0

iii)  $\lim_{x \to\ 0}$ [sin(x) / x] = 1

Using the definition of a derivative:

f’(x) =  $\lim_{h \to\ 0}$[{f(x+h) – f(x)} / h]  

Now, by substituting cos x for our function,

cos’ (x)  

=  $\lim_{h \to\ 0}$[{cos(x+h) – cos(x)} / h]

= $\lim_{h \to\ 0}$[{(cosx cosh – sinx sinh) – cos(x)} / h] ....... [using formula (i)]

taking cos x as a factor

= $\lim_{h \to\ 0}$[{cosx (cosh – 1) - sinx sinh} / h]

=  $\lim_{h \to\ 0}$[cosx {(cosh – 1) / h} - sinx {sinh / h}]

using formula (ii) & (iii)

= $\lim_{h \to\ 0}$[(cos x * 0) - sinx * 1]

= $\lim_{h \to\ 0}$[- sinx]

since there are no more h variables therefore  $\lim_{h \to\ 0}$ can be neglected  

= - sin(x)