From what you know about the graphs of functions (i) through (v), pick out the one that is the derivative of f for x>0.
Show your work.
f(x)=x ln x-x
i. y=sin x
ii. y=ln x
iii. y= x^(1/2)
iv. y=x^2
v. y=3x-1
f(x)=x ln x-x
f'(x) = x(1/x) + lnx - 1
= 1 + lnx - 1
= lnx
To find the derivative of the function f(x), we can use the rules of differentiation. The derivative of f(x) can be written as f'(x).
f(x) = x ln(x) - x
To find the derivative, we will differentiate each term separately using the rules of differentiation.
Derivative of x ln(x):
For the term x ln(x), we can use the product rule. The product rule states that if we have a function u(x) multiplied by another function v(x), then the derivative of the product is given by:
(uv)' = u'v + uv'
Let's apply the product rule to x ln(x):
u(x) = x
v(x) = ln(x)
u'(x) = 1 (since the derivative of x with respect to x is 1)
v'(x) = 1/x (since the derivative of ln(x) with respect to x is 1/x)
Using the product rule, we have:
(x ln(x))' = u'v + uv'
= (1)(ln(x)) + (x)(1/x)
= ln(x) + 1
Derivative of -x:
The derivative of a constant multiple x is simply -1.
Therefore, f'(x) = ln(x) + 1 - 1
= ln(x)
From the given options (i) through (v), the function that matches the derivative of f(x) for x > 0 is ii. y = ln(x).
To find the graph that represents the derivative of the function f(x) for x > 0, let's first recall the definition of a derivative. The derivative of a function f(x) at a specific point x is the slope of the tangent line to the graph of f(x) at that point.
In this case, we need to find the derivative of f(x) = x ln x - x and determine which graph best represents it for x > 0.
To find the derivative, we can use the rules of differentiation. We'll differentiate each term of the function separately and then combine the results.
1. Differentiating x ln x:
The derivative of x ln x can be found using the product rule.
Let u = x and v = ln x. By the product rule:
d(uv)/dx = u(dv/dx) + v(du/dx)
= x(1/x) + ln x(1)
= 1 + ln x
2. Differentiating -x:
The derivative of -x is -1.
Now, adding the results from steps 1 and 2, we have the derivative of f(x):
f'(x) = 1 + ln x - 1
= ln x
Therefore, the derivative of f(x) is ln x.
From the given options, the graph that represents the function ln x for x > 0 is (ii) y = ln x.