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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)
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