Differentiate the function with respect to x. sin(x²+ 5)

Solution:

Let f(x) = sin (x²+ 5),

u(x) = x²+ 5

f(x) = sin⁡ (x²+ 5),

u(x) = x²+ 5 and v(t) = sin t

Then, (vou)(x) = v(u(x)) = v(x²+ 5)

= tan (x²+ 5) = f(x)

Thus, f is a composite of two functions.

Put, t = u(x) = x²+ 5t

Then, we get

dv/dt = d/dt(sint)

= cos t = cos (x²+ 5) dt / dx

= d/dx (x²+ 5)

= d/dx (x²) + d / dx (5)

= 2x + 0 = 2x

By chain rule of derivative,

df / dx = dv / dt. dt / dx = cos (x²+ 5) × 2x = 2x cos (x²+ 5)

Alternate method:

d/dx [sin (x²+ 5)] = cos (x²+ 5).d / dx (x²+ 5)

= cos (x²+ 5).[d / dx (x²) + d / dx (5)]

= cos (x²+ 5).[2x + 0]

= 2x cos (x²+ 5)

NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.2 Question 1