Problem

Differentiate. \[ f(x)=x^{6} \ln 5 x \]

Solution

Step 1 :The given function is a product of two functions, \(x^6\) and \(\ln(5x)\).

Step 2 :To differentiate this function, we will use the product rule of differentiation which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Step 3 :The derivative of \(x^6\) is \(6x^5\).

Step 4 :The derivative of \(\ln(5x)\) is \(\frac{1}{x}\).

Step 5 :So, the derivative of the given function is \(6x^5 \ln(5x) + x^6 \times \frac{1}{x}\).

Step 6 :Simplifying the expression, we get \(6x^5 \ln(5x) + x^5\).

Step 7 :Final Answer: The derivative of the function \(f(x)=x^{6} \ln 5 x\) is \(\boxed{6x^{5} \ln(5x) + x^{5}}\).

From Solvely APP
Source: https://solvelyapp.com/problems/8837/

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