Differentiate the function with respect to x.  cos(√x)

Solution:

A derivative helps us to know the changing relationship between two variables. Consider the independent variable 'x' and the dependent variable 'y'. 

Let f(x) = cos (√x)

Also, let u(x) = √x and,

v(t) = cos t

Then, (vou) (x) = v (u (x))

= v(√x) = cos √x = f(x)

Since, f is a composite function of u and v.

t = u(x) = √x

Then,

dt/dx = d/dx (√x)

= d/dx (x^1/2) = 1/2 x ^−1/2

= 1/2√x

And,

dv/dt = d/dt (cos t)

= − sin t = −sin (√x)

Using chain rule, we get

dt/dx = dv/dt. dt/dx

= −sin (√x).1/2 √x

= −1/2 √x sin (√x)

= −sin (√x) / 2√x

Alternate method:

d/dx [cos (√x)] = −sin (√x).d/dx (√x)

= − sin (√x) × d/dx (x ^1/2)

= − sin √x × 1 / 2x ^{− 1/2}

⇒ d/dx (cos(√x)) = − sin √x / 2√x

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.2 Question 8

Differentiate the function with respect to x. cos(√x)

Summary:

The derivative of the function with respect to x, cos(√x) is d/dx (cos(√x)) = − sin √x / 2√x