Find the derivative of the function. HINT [See Quick Examples on page 835.]
f(x) = 9x ln x
f'(x)=9[1+lnx]
9x(1/x) + lnx (9)
9 + 9 ln x
9(1+lnx) agreed
To find the derivative of the function f(x) = 9x ln x, we can use the product rule.
The product rule states that if we have two functions u(x) and v(x), then the derivative of the product u(x) * v(x) is given by the formula:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
In our case, u(x) = 9x and v(x) = ln x. Let's find the derivatives of these functions first.
To find the derivative of u(x) = 9x, we can use the power rule. The power rule states that if we have a function of the form f(x) = cx^n, then its derivative is given by:
d/dx(cx^n) = c * n * x^(n-1)
Applying this rule, we find that the derivative of u(x) = 9x is:
u'(x) = 9 * 1 * x^(1-1) = 9
To find the derivative of v(x) = ln x, we can use the chain rule. The chain rule states that if we have a function of the form f(g(x)), then its derivative is given by:
d/dx(f(g(x))) = f'(g(x)) * g'(x)
In our case, f(x) = ln x and g(x) = x. The derivative of f(x) = ln x, also denoted as (d/dx)(ln x), is a well-known derivative and can be found using calculus identities. The derivative of ln x is 1/x.
Applying the chain rule, we find that the derivative of v(x) = ln x is:
v'(x) = (d/dx)(ln x) * (d/dx)(x) = (1/x) * 1 = 1/x
Now that we have the derivatives of u(x) and v(x), we can use the product rule to find the derivative of f(x):
f'(x) = u'(x) * v(x) + u(x) * v'(x)
= 9 * ln x + 9x * (1/x)
= 9 ln x + 9
Therefore, the derivative of the function f(x) = 9x ln x is f'(x) = 9 ln x + 9.