Question

Find the derivative of the following expression: y = 2x^3 - 5x / (x^2 - 3x^4).

Answer 1

The problem involves finding the derivative of y = 2x^3 - 5x / (x^2 - 3x^4), which is a calculus problem requiring the application of the quotient rule and the power rule.

Step-by-step explanation:

The student has asked for help in finding the derivative of the function y = 2x^3 - 5x / (x^2 - 3x^4). This is a problem of differentiation which can be solved using the quotient rule and power rule in calculus. The quotient rule states that the derivative of a function in the form of f(x)/g(x) is (g(x)*f'(x) - f(x)*g'(x)) / [g(x)]^2. The power rule states that the derivative of x^n is n*x^(n-1).

Here is the step-by-step differentiation:

1. Identify the top function (numerator) and the bottom function (denominator). In this case, the top function is f(x) = 2x^3 - 5x, and the bottom function is g(x) = x^2 - 3x^4.
2. Find the derivatives of both f(x) and g(x). f'(x) = 6x^2 - 5 and g'(x) = 2x - 12x^3.
3. Apply the quotient rule: y' = (g(x)*f'(x) - f(x)*g'(x)) / [g(x)]^2 = [(x^2 - 3x^4)(6x^2 - 5) - (2x^3 - 5x)(2x - 12x^3)] / (x^2 - 3x^4)^2.
4. Simplify the expression to complete the differentiation.

The final result after simplification will yield the derivative of y with respect to x.