Derivative of xlnx
The derivative of xlnx is equal to ln x + 1 and it is given by the process of differentiation of xlnx. It can be calculated using the product rule of differentiation. The formula for the derivative of xlnx is mathematically written as d(xlnx)/dx OR (xlnx)' = lnx + 1. We can also evaluate the derivative of xlnx using the first principle of derivatives, that is, the definition of limits. The differentiation of a function gives the rate of change in the function with respect to the variable.
In this article, we will calculate the derivative of x lnx using the first principle of differentiation and the product rule of derivatives and, hence derive its formula. We will also solve some examples using the derivative of xlnx to understand the concept better.
1.  What is Derivative of xlnx? 
2.  Derivative of xlnx Formula 
3.  Derivative of xlnx By First Principle 
4.  Derivative of xlnx By Product Rule 
5.  FAQs on Derivative of xlnx 
What is Derivative of xlnx?
The derivative of xlnx gives the rate of change in the function f(x) = xlnx with respect to the variable x. It can be evaluated using different methods of differentiation including the first principle (definition of limits) and the product rule of differentiation. The derivative of xlnx is equal to lnx + 1. To evaluate this derivative using the product rule, we can consider x as the first function and lnx as the second function as xlnx is the product of x and lnx and the formula for the derivative of x and the derivative of lnx. Let us go through the formula of the derivative of xlnx in the next section.
Derivative of xlnx Formula
The formula for the derivative of xlnx can be written in the following two ways:
 d(xlnx)/dx = ln x + 1
 (xlnx)' = ln x + 1
Let us now prove these formulas using various methods of differentiation.
Derivative of xlnx By First Principle
In this section, we will determine the derivative of xlnx using the first principle of derivatives, that is, the definition of limits. To derive the derivative of f(x) = xlnx, we take the limiting value as x approaches x + h. To simplify this, we set x = x + h, and we want to take the limiting value as h approaches 0. We will use the following formulas to prove the result:
 f'(x) = lim_{h→0}[f(x+h)  f(x)]/[(x+h)  x]
 lim_{x→0} [ ln (1 + x) ] / x = 1
 ln a  ln b = ln(a/b)
Using the above formulas, we have
d(xlnx)/dx = lim_{h→0}[(x+h) ln(x+h)  xlnx]/[(x+h)  x]
= lim_{h→0}[x ln(x+h) + h ln(x+h)  xlnx]/h
= lim_{h→0}[x ln(x+h)  x lnx + h ln(x+h)]/h
= lim_{h→0}[x(ln(x+h)  lnx) + h ln(x+h)]/h
= lim_{h→0}[x ln [(x+h)/x] + h ln(x+h)]/h
= lim_{h→0}[x ln (1+h/x) + h ln(x+h)]/h
= lim_{h→0} [x ln (1+h/x)]/h + lim_{h→0} ln(x+h)
= lim_{h→0} [ ln (1 + h/x) ] / (h/x) + lim_{h→0} ln(x+h)
= 1 + lnx  [Using limit formula lim_{x→0} [ ln (1 + x) ] / x = 1]
Hence, we have proved that the formula for the derivative of xlnx is equal to 1 + lnx.
Derivative of xlnx By Product Rule
Now that we know that the derivative of xlnx is equal to 1 + lnx, we will prove it using the product rule of differentiation. According to the product rule, the derivative of function h(x) = f(x) g(x) is given by, h'(x) = f'(x) g(x) + f(x) g'(x). For h(x) = xlnx, we have f(x) = x and g(x) = ln x. Also, we know that the derivative of x is equal to 1 and the derivative of ln x is equal to 1/x. Using these formulas, we have
d(xlnx)/dx = (x)' × lnx + x × (ln x)'
= 1 × lnx + x × (1/x)
= lnx + 1
Hence, we have derived the formula for the derivative of xlnx using the product rule.
Important Notes on Derivative of xlnx
 The formula for the derivative of xlnx is given by 1 + lnx.
 We can evaluate the differentiation of xlnx using the first principle and product rule methods of differentiation.
 We use the formulas of the derivative of x, the derivative of lnx, and the limit formulas to find the derivative of xlnx.
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Examples Using Derivative of xlnx

Example 1: What is the second derivative of xlnx?
Solution: To find the second derivative of xlnx, we will differentiate the first derivative of xlnx. As we know the derivative of xlnx is equal to 1 + lnx, we have
d^{2}(xlnx)/dx^{2} = d(xlnx)/dx
= d(1 + lnx)/dx
= d(1)/dx + d(lnx)/dx  [Using derivative of sum of functions formula: d(u+v)/dx = du/dx + dv/dx]
= 0 + 1/x  [Because the derivative of a constant function is zero and the derivative of lnx is equal to 1/x.]
= 1/x
Answer: The second derivative of xlnx is 1/x.

Example 2: Find the derivative of x lnx  x.
Solution: To evaluate the derivative of xlnx  x, we will use the formula for the derivative of difference of functions given by, d(uv)/dx = du/dx  dv/dx. Using this, we have
d(xlnx  x)/dx = d(xlnx)/dx  dx/dx
= 1 + lnx  1  [Because derivative of xlnx is equal to 1 + lnx and the derivative of x is 1]
= lnx
Answer: The derivative of x lnx  x is lnx.
FAQs on Derivative of xlnx
What is the Derivative of xlnx in Calculus?
The derivative of xlnx is equal to ln x + 1. It can be evaluated using different methods of differentiation including the first principle of derivatives and the product rule of differentiation.
What is the Formula for the Derivative of xlnx?
The formula for the derivative of xlnx is given by, d(xlnx)/dx OR (xlnx)' = lnx + 1. We can derive this formula using the first principle of derivatives and the product rule method of differentiation.
How to Find the Derivative of xlnx?
We can find the derivative of xlnx using various methods of differentiation including the product rule and the first principle of differentiation. We can use the formulas for the derivatives of x and lnx, and formulas of limits.
What is the Derivative of xlnx  x?
The derivative of xlnx  x is equal to lnx. It can be calculated using the formulas of the derivative of xlnx and the derivative of x. We can apply the formula for the derivative of difference of functions given by, d(uv)/dx = du/dx  dv/dx.
How to Find the Second Derivative of xlnx?
We can find the second derivative of xlnx by differenting the first derivative of xlnx. We can simply differentiation d(xlnx)/dx = 1 + lnx to obtain the second derivative of xlnx.
What is the Derivative of e^{x} lnx?
The derivative of e^{x} lnx is equal to e^{x}(lnx + 1/x). It is calculated as d(e^{x} lnx)/dx = (e^{x})' lnx + e^{x} (lnx)' = e^{x} lnx + e^{x} (1/x) = e^{x} (lnx + 1/x) using the product rule method of differentiation.
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