Final answer:
To find the derivative of the function y=x³+2/(4x+1)(2x+3), we use the power rule for the first term and the quotient rule for the second term. The derivative is y′ = 3x² - (32x + 28)/((4x+1)^2•(2x+3)^2).
Step-by-step explanation:
Find the Derivative of the Function y=x³+2/(4x+1)(2x+3)
To find the derivative of the function y=x³+2/(4x+1)(2x+3), we need to apply the rules of differentiation.
The function can be seen as the sum of two terms, x³ and 2/(4x+1)(2x+3). The derivative of the first term, x³, is straightforward and can be found using the power rule:
y′ = 3x²
For the second term, we need to use the quotient rule since it is a ratio of two functions. The quotient rule states that if we have a function that is the quotient of two functions, u(x)/v(x), its derivative is:
y′ = (v•u′ - u•v′)/v²
Applying the quotient rule to 2/(4x+1)(2x+3), let's find the derivatives of the numerator (which is simply 0 since the numerator is a constant) and the denominator (applying the product rule).
The derivative of the denominator is:
v′ = (4x+1)′•(2x+3) + (4x+1)•(2x+3)′
v′ = 4•(2x+3) + (4x+1)•2
v′ = 8x + 12 + 8x + 2
v′ = 16x + 14
So the derivative of the second term is:
•y′ = (0 - 2•(16x+14))/((4x+1)^2•(2x+3)^2)
•y′ = -32x - 28
Combining both terms, the complete derivative of the function y is:
y′ = 3x² - (32x + 28)/((4x+1)^2•(2x+3)^2)