Question

Question:

if f(x)=x cos x, find f"(x), or d2ydx2

Physics Calculus Differentiation

Solution:

Using the Product Rule, we have f'(x)=xddx(cos x)+cos xddx(x)=-xsinx+cosx
To find f"(x) we differentiate f'(x):
f"(x)=ddx(-x sin x+cos x)=-xddx(sin x)+sin x ddx(-x)+ddx(cos x)
=-x cos x-sin x-sin x = -x cos x-2 sin x

my problem is :

I didn't understand how to solve this question

Answer 1

y= x cos x

product rule says that
Derivative of f(x)×g(x) is equal to f(x)×derivative of g(x)+g(x)×derivative of f(x)

this is applied in the solution

derivative of sinx is cosx and that of cosx is -sinx

=> dy/dx
= x * d/dx (cos x) + (cos x) * d/dx (x)
= - x sin x + cos x

d^2y/dx^2
= d/dx (dy/dx)
= d/dx ( - x sin x + cos x)
= - x * d/dx (sin x) - sin x * d/dx (x) + d/dx (cos x)
= - x cos x - sin x - sin x
= - x cos x - 2 sin x