Question

How do you differentiate \[f(x)=2x\ln x\] using product rule?

Answer 1

Hint: The product rule is a rule for differentiating expressions in which one function is multiplied by another function. The rule follows from the limit definition of derivative and is given by \[\dfrac{dy}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\] if \[y=uv\] where \[\dfrac{dy}{dx}\] means derivative of y with respect to x, \[\dfrac{dv}{dx}\] means derivative of v with respect to x and \[\dfrac{du}{dx}\] means derivative of u with respect to x. The above formula is applicable only if u and v are differentiable.

Complete step by step answer:
As per the given question, we have to differentiate the given function using the product rule. Here, the given function to be differentiated is \[f(x)=2x\ln x\].
Now, let \[y=f(x)\] then \[u=2x\] and \[v=\ln x\]. We know that, according to power rule,
derivative of \[a{{x}^{n}}\] is given by \[\dfrac{d}{dx}(a{{x}^{n}})=(na){{x}^{n-1}}\] and the derivative of logarithmic function is given by \[\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}\]. Then,
The derivative of function u: -
By comparing the function ‘u’ with the function of power rule, we get the coefficient a=2 and power of x, n=1. Then,
\[\Rightarrow \dfrac{du}{dx}=2\] \[(\Leftarrow \]according to power rule)
The derivative of function v: -
By comparing the function ‘v’ with the function of logarithmic rule, function ‘v’ is the same as the logarithmic function. Then,
\[\Rightarrow \dfrac{dv}{dx}=\dfrac{1}{x}\] (from derivative of logarithmic function)

Now substituting the above derivatives in product rule, we get
\[\Rightarrow \] \[\dfrac{dy}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\]
\[\Rightarrow \]\[\dfrac{dy}{dx}=(2x)\dfrac{1}{x}+(\ln x)2\]
On solving the above equation, we get
\[\Rightarrow \dfrac{dy}{dx}=\dfrac{d(f(x))}{dx}=2+2(\ln x)=2(1+\ln x)\]
\[\therefore \] The derivative of the given function \[f(x)=2x\ln x\] is \[2(1+\ln x)\].

Note:
In order to solve these types of problems, we must have enough knowledge about derivatives of basic functions. The product rule is nothing but take the derivative of v multiplied by u and add v multiplied by the derivative of u. We have the product rule for 3 functions also which is given by \[\dfrac{dy}{dx}=\dfrac{df(x)}{dx}g(x)h(x)+f(x)\dfrac{dg\left( x \right)}{dx}h(x)+f(x)g(x)\dfrac{dh(x)}{dx}\] if \[y=f(x)g(x)h(x)\].