Find the derivative of the given function.
f(x)=(�ãx+9x)(x^3/2-x)
f '(x) = ??
Evaluate f '(x) at the value x = 4.
f '(4) = ??
To find the derivative of the given function f(x) = (x+9x)(x^(3/2)-x), we can apply the product rule of differentiation, which states that if u(x) and v(x) are differentiable functions with respect to x, then the derivative of the product of u(x) and v(x) is given by:
[d(uv(x))/dx] = u(x)*[dv(x)/dx] + v(x)*[du(x)/dx]
Now, let's break down the given function f(x) into two separate functions u(x) and v(x):
u(x) = (x + 9x)
v(x) = (x^(3/2) - x)
Next, let's find the derivatives of u(x) and v(x):
[u'(x) = ] [d/dx] (x + 9x)
= 1 + 9 (derivative of x is 1 and 9x is 9)
[v'(x) = ] [d/dx] (x^(3/2) - x)
= (3/2)x^(1/2) - 1 (using the power rule of differentiation: d/dx (x^n) = nx^(n-1))
Now, we can substitute these values into the product rule:
f'(x) = u(x)*v'(x) + v(x)*u'(x)
= (x + 9x)*[(3/2)x^(1/2) - 1] + (x^(3/2) - x)*[1 + 9]
Simplifying this expression will give us the derivative f'(x) of the given function.
To evaluate f'(x) at x = 4, we substitute x = 4 into the expression for f'(x):
f'(4) = (4 + 9*4)*[(3/2)*4^(1/2) - 1] + (4^(3/2) - 4)*[1 + 9]
By simplifying this expression, we can find the value of f'(4).