Question

Find the derivative of the function: y = ln(5x^2-x) + 4x

Answer 1

The derivative of the function y = ln(5x^2-x) + 4x is found using the chain and power rules, resulting in (10x - 1)/(5x^2 - x) + 4.

Step-by-step explanation:

To find the derivative of the function y = ln(5x^2-x) + 4x, we'll use the rules of differentiation, specifically the chain rule for natural logarithms and the power rule.

Step by step solution:

1. Let's first differentiate ln(5x^2-x). By the chain rule:
2. Derivative of ln(u) is 1/u, here u = 5x^2 - x. So we get 1/(5x^2 - x).
3. Next, we need the derivative of u, which is the inside function 5x^2 - x. This is 10x - 1 by the power rule.
4. Now, multiply the derivatives: 1/(5x^2 - x) * (10x - 1) to get (10x - 1)/(5x^2 - x).
5. Finally, the differentiation of 4x simply gives us 4.
6. Add these results to find the derivative of the whole function: d/dx[ln(5x^2 - x)] + d/dx[4x] = (10x - 1)/(5x^2 - x) + 4.

So, the derivative of y = ln(5x^2-x) + 4x is (10x - 1)/(5x^2 - x) + 4.