Question

Find the derivative of the following functions. $$y=x \ln x-x$$

Short answer

Answer: The derivative of the function $$y=x \ln x-x$$ is $$y'=\ln x$$.

Step-by-step solution

Identify Terms to Derive Individually

Since we have the function $$y=x \ln x-x$$, we need to first separate the equation into two terms. We can rewrite this as $$y=(x \ln x) - x$$ and derive them individually.

Apply the Sum/Difference Rule

According to the sum/difference rule, we can find the derivative of the function by finding the derivative of each term separately and then combining the results. So, we need to compute $$\frac{d}{dx}(x \ln x)$$ and $$\frac{d}{dx}(-x)$$.

Derive the Second Term using Constant Rule

To derive the second term, $$-x$$, we apply the constant rule, which states that the derivative of any constant times a variable is just the constant. Therefore, $$\frac{d}{dx}(-x)=-1$$.

Derive the First Term using Product Rule

To find the derivative of the first term, $$x\ln x$$, we need to apply the product rule since it is a product of two functions: $$u(x)=x$$ and $$v(x)=\ln x$$. The product rule states that $$(uv)'=u'v+uv'$$. So, we first compute $$u'(x)$$ and $$v'(x)$$. We already know that $$u'(x)=1$$, as calculated in Step 3. To compute $$v'(x)$$, remember that the derivative of $$\ln x$$ is $$\frac{1}{x}$$. Hence, $$v'(x)=\frac{1}{x}$$. Now, we apply the product rule: $$\frac{d}{dx}(x \ln x) = (1) (\ln x) + (x) \left(\frac{1}{x} \right)$$.

Simplify the expression

Next, we simplify the expression from Step 4: $$\frac{d}{dx}(x \ln x) = \ln x + \cancel{x} \left(\frac{1}{\cancel{x}} \right) = \ln x + 1$$.

Combine the Results

Finally, we combine the derivatives of the two terms, obtained in Steps 3 and 5: $$y' = \frac{dy}{dx} = (\ln x + 1) - 1 = \ln x$$. Therefore, the derivative of the function $$y=x \ln x-x$$ is $$\boxed{y'=\ln x}$$.

Key concepts

Product Rule

In calculus, the product rule is a fundamental tool that allows us to find the derivative of the product of two functions. It states that for two differentiable functions, say \( u(x) \) and \( v(x) \) the derivative of the product \( uv \) is given by \( (uv)' = u'v + uv' \).

When finding the derivative of a function like \( x\ln(x) \) as in our exercise, we treat each part of the product—\( x \) and \( \ln(x) \)—as separate functions (\( u \) and \( v \) respectively). We then differentiate each one (\( u'(x) = 1 \) and \( v'(x) = \frac{1}{x} \) for the natural logarithm of \( x \) and combine them according to the product rule formula. The differentiated terms are then added together, yielding \( \ln x + 1 \), where the \( 1 \) comes from \( x \) being multiplied by its derivative which is \( \frac{1}{x} \).

Constant Rule

The constant rule in differentiation states that the derivative of a constant multiplied by a variable is simply the constant itself. This means for any constant \( a \) and variable \( x \) the derivative of \( ax \) with respect to \( x \) is \( a \).

In our example \( -x \), we treat \( -1 \) as the constant \( a \) and so the derivative becomes \( -1 \), since the variable \( x \) itself has a derivative of \( 1 \) and \( -1 \times 1 = -1 \). This rule demonstrates the simplification power in calculus, especially when dealing with terms that can be quickly dispatched without much computation.

Sum/Difference Rule

The sum/difference rule instructs us on how to handle derivatives of functions that are being added or subtracted. It simply states that the derivative of a sum or difference is the sum or difference of the derivatives. Symbolically, for two functions \( u(x) \) and \( v(x), \) the rule is written as \( (u \pm v)' = u' \pm v' \).

This rule is particularly sleek because it allows us to break down more complex equations into their constituent parts, differentiating each part separately. This was utilized in our exercise where we split the function \( y = x\ln x - x \) and applied the rule to find \( (x\ln x)' - (x)' \) before simplifying and recombining them, aiding in the extraction of the solution \( y' = \ln x \).

Natural Logarithm Differentiation

Differentiation of the natural logarithm function, \( \ln x \), is a specific instance where a predefined derivative is applied. The derivative of \( \ln x \) is \( \frac{1}{x} \) on the domain of \( x > 0 \).

This rule is essential for functions involving \( \ln x \), like the product \( x\ln x \) seen in our exercise. Upon encountering \( \ln x \) while differentiating, one transitions immediately to using \( \frac{1}{x} \) as its derivative without further analysis, expediting the process. This is a powerful shortcut for students as they familiarize themselves with the common derivatives of fundamental functions.