Question

How do you find the derivative of $ y=x{{\ln }^{3}}x $ ?

Answer 1

Hint: In this question, we have to find the derivative of an equation. A derivative is the rate of change function with respect to its variable. Thus, in this question, we have to find a simpler form that is developed from this equation. So, to solve this problem, we will use the product rule method. We first rewrite the given equation in the form $ y=u.v $ and then find the individual derivatives of the terms. After that, we will put the value in the equation $ \dfrac{dy}{dx}=\dfrac{du}{dx}.v+u.\dfrac{dv}{dx} $ and solve it further, which is the required solution to the problem.

Complete step by step answer:
According to the question, we have to find the derivative of an equation.
Thus, we will use the product rule method to solve the same.
The equation given to us is $ y=x{{\ln }^{3}}x $ ---------- (1)
We know that the RHS of the function consists of two-term which is multiplied by each other, that is one is $ x $ and another is $ {{\ln }^{3}}x $ .
Thus, we first apply the product rule, which states that
If $ y=u.v $ then ------------ (2)
 $ \dfrac{dy}{dx}=\dfrac{du}{dx}.v+u.\dfrac{dv}{dx} $ ----------- (3)
Therefore, on comparing equations (1) and (2), we get that
 $ u=x $ and ------------- (4)
 $ v={{\ln }^{3}}x $ --------- (5)
So, we first find the derivative of equation (4), that is, the derivative of u with respect to x, we get
 $ \Rightarrow \dfrac{du}{dx}=1 $ -------------- (6)
Now, we will find the derivative of equation (5), that is we will apply the formula $ {{\left( {{x}^{a}} \right)}^{\prime }}=a.\left( {{x}^{a-1}} \right) $ in the equation (5), we get
 $ \Rightarrow \dfrac{dv}{dx}=3.({{\ln }^{2}}x).\left( \dfrac{1}{x} \right) $ --------- (7)
Now, we will put the value of equation (4), (5), (6), and (7) in equation (3), we get
 $ \Rightarrow \dfrac{dy}{dx}=(1).\left( {{\ln }^{3}}x \right)+(x).\left( 3.({{\ln }^{2}}x).\left( \dfrac{1}{x} \right) \right) $
On further solving, we get
 $ \Rightarrow \dfrac{dy}{dx}=\left( {{\ln }^{3}}x \right)+\left( 3.({{\ln }^{2}}x) \right) $
Now, we will take $ {{\ln }^{2}}x $ common in the above equation, we get
 $ \Rightarrow \dfrac{dy}{dx}={{\ln }^{2}}x\left( \ln x+3 \right) $ which is the required solution to the problem.
Therefore, for the equation $ y=x{{\ln }^{3}}x $ , its derivative value is $ {{\ln }^{2}}x\left( \ln x+3 \right) $.

Note:
  While solving this problem, mention all the identities and the formula you are using to avoid mathematical errors and to get an accurate answer. While solving the derivative of $ v={{\ln }^{3}}x $, we first solve the exponent of the function, and then solve the given function, to get the correct answer.