Final answer:
The derivative of f(x) = xcosx is found by applying the product rule, resulting in f'(x) = cos(x) - xsin(x). The correct answer is option B .
Step-by-step explanation:
The question 'Find the differential of f(x) = xcosx' refers to finding the derivative of the given function with respect to x. Since the function consists of two functions of x multiplied together (that is, x and cosx), we need to apply the product rule for differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function.
Applying this rule to f(x) = xcosx, we get:
f'(x) = d/dx(x) · cosx + x · d/dx(cosx)f'(x) = 1 · cosx + x · (-sinx)f'(x) = cosx - xsinx
Therefore, the correct differential of the function f(x) = xcosx is given by option b, which is f'(x) = cos(x) - xsin(x).
To find the differential of the function f(x) = xcosx, we need to use the product rule of differentiation. The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by:
f'(x) = u'(x)v(x) + u(x)v'(x)
In this case, u(x) = x and v(x) = cosx. The derivative of u(x) is 1, and the derivative of v(x) is -sinx. Plugging these values into the product rule formula, we get:
f'(x) = 1*cosx + x*(-sinx) = cosx - xsinx
Therefore, the correct answer is b) f'(x) = cos(x) - xsin(x).