Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): sin (x + a)

Solution:

We know that d/dx (sin x) = cos x.

But here we have to find the derivative of sin (x + a). So the above formula directly cannot be applied here. So we will use the first principles to find the derivative.

The given function is f(x) = sin (x + a). Its derivative is,

f' (x) = limₕ→₀ [f (x + h) - f (x)]/h

= limₕ→₀ [ sin (x+h+a) - sin (x+a) ] / h

We know that sin A - sin B = 2 cos (A+B)/2 sin (A-B)/2

=  limₕ→₀ [2 cos (x+h+a+x+a)/2 sin (x+h+a-x-a)/2 ] / h

=  limₕ→₀ [ 2 cos ((2x+h+2a)/2) sin (h/2) ] / h

Multiply and divide by 1/2, 

=  limₕ→₀ [ 2 cos ((2x+h+2a)/2) ] [ (1/2) limₕ/₂→₀ sin (h/2) / (h/2)]

= [2 cos ((2x+2a)/2)] [(1/2) (1)]   (As limₓ→₀  sin x / x = 1)

= cos (x + a)

NCERT Solutions Class aa Maths Chapter 13 Exercise ME Question 14

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): sin (x + a)

Summary:

The derivative of the given function is sin (x + a) is cos (x + a)