Question

Find the derivative of the function.
y=x³+2/(4 x+1)(2 x+3)
yʹ=___

Answer 1

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)