# How do I differentiate (2x+1) / (3x^2 - 5)?

This is a typical example where the quotient rule is required to answer the question. This is clear because the function is made up of two other functions of x, one is the numerator and the other is the denominator. Also, the function is not easily simplifiable.

Therefore, you must separate the functions into two new ones, let's call them function M and function N, where M is 2x+1 and N is 3x^2 - 5. We then differentiate each function as we would normally. M differentiates to give 2, and N gives 6x. We then plug these expressions into the quotient rule and simplify.

This should give (2(3x^2 - 5) - 6x(2x+1)) / (3x^2 - 5)^2, which simplifies to (-6x^2 -6x - 10) / (3x^2 - 5)^2. We can see that this expression cannot be simplified further because when we use the quadratic formula to factorise the numerator, the part inside the square root is negative, therefore we cannot simplify this expression and this is the final answer.

Answered by Sophie T. Maths tutor

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