Find the derivative of each of the following functions using the product rule
a. f(x) =x sin x
To find the derivative of f(x) = x sin x using the product rule, we differentiate each function and apply the product rule.
Let u = x and v = sin x.
Using the product rule, the derivative of f(x) is given by:
f'(x) = u'v + uv'
Differentiating u = x, we have:
u' = 1
Differentiating v = sin x, we have:
v' = cos x
Substituting these values into the product rule formula, we get:
f'(x) = (1)(sin x) + (x)(cos x)
f'(x) = sin x + x cos x
Therefore, the derivative of f(x) = x sin x is f'(x) = sin x + x cos x.
To find the derivative of f(x) = x sin(x) using the product rule, we need to differentiate each part of the product separately.
Let's break down f(x) = x sin(x) into two parts:
u(x) = x
v(x) = sin(x)
Using the product rule, the derivative of f(x) is given by:
f'(x) = u'(x)v(x) + u(x)v'(x)
First, let's find u'(x) (the derivative of u(x)):
u'(x) = 1 (since the derivative of x is 1)
Next, let's find v'(x) (the derivative of v(x)):
v'(x) = cos(x) (since the derivative of sin(x) is cos(x))
Now we can substitute these values into the product rule formula:
f'(x) = u'(x)v(x) + u(x)v'(x)
= (1)(sin(x)) + (x)(cos(x))
= sin(x) + xcos(x)
Therefore, the derivative of f(x) = x sin(x) is f'(x) = sin(x) + xcos(x).