find the derivative of the function f(x) = x^2 cos (7x)
To find the derivative of the function f(x) = x^2 * cos(7x), you can use the product rule of differentiation. The product rule states that if you have two functions u(x) and v(x), the derivative of their product is given by:
(d/dx) [u(x) * v(x)] = u'(x) * v(x) + u(x) * v'(x)
In this case, u(x) = x^2 and v(x) = cos(7x). Let's find their derivatives first:
u'(x) = 2x (Applying the power rule for differentiation)
v'(x) = -7 * sin(7x) (Applying the chain rule for differentiation)
Now, using the product rule, we can find the derivative of f(x):
f'(x) = u'(x) * v(x) + u(x) * v'(x)
= (2x) * cos(7x) + (x^2) * (-7 * sin(7x))
Simplifying this expression gives us the derivative of f(x):
f'(x) = 2x * cos(7x) - 7x^2 * sin(7x)
Therefore, the derivative of the function f(x) = x^2 * cos(7x) is f'(x) = 2x * cos(7x) - 7x^2 * sin(7x).
use the product rule and the chain rule to get
2x cos(7x) - 14x^2 sin(7x)