Find the derivative of the following functions (it is to be understood that a is a fixed non-zero constant): sin (x + a) / cos x

Solution:

Finding the derivative of sin (x + a):

First, let us find the derivative of sin (x + a). Let us assume that f(x) = sin (x + a). 

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)

Finding the derivative of the given function:

The given function is g(x) = sin (x + a) / cos x.

Its derivative using the quotient rule is,

d/dx (g(x)) = [ (cos x) d/dx (sin (x + a)) - (sin (x + a)) d/dx (cos x) ] / (cos x)²

= [ (cos x) (cos (x + a)) - (sin (x + a)) (- sin x) ] / cos²x (using derivative formulas)

= [ (cos x) (cos (x + a)) + (sin (x + a)) (sin x) ] / cos²x

We know that cos A cos B + sin A sin B = cos (A - B). Using this,

d/dx (g(x)) = [ cos (x - (x + a)) ] / cos²x

= [ cos (x - x - a) ] / cos²x

= cos (-a) / cos²x

= cos a / cos²x    (as cos (-a) = cos a)

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NCERT Solutions Class 11 Maths Chapter 13 Exercise ME Question 21

Find the derivative of the following functions (it is to be understood that a is a fixed non-zero constant): sin (x + a) / cos x

Summary:

The derivative of the given function sin (x + a) / cos x is cos a / cos²x