Questions: Find the derivative of the function. y=(7x^2+2)/(x^2+3)

Solution Steps

To find the derivative of the given function, we will use the quotient rule. The quotient rule states that if you have a function \( y = \frac{u(x)}{v(x)} \), then its derivative \( y' \) is given by \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Here, \( u(x) = 7x^2 + 2 \) and \( v(x) = x^2 + 3 \). We will first find the derivatives \( u'(x) \) and \( v'(x) \), and then apply the quotient rule.

The given function is \( y = \frac{7x^2 + 2}{x^2 + 3} \). We identify the numerator as \( u(x) = 7x^2 + 2 \) and the denominator as \( v(x) = x^2 + 3 \).

Calculate the derivatives of \( u(x) \) and \( v(x) \):

• \( u'(x) = \frac{d}{dx}(7x^2 + 2) = 14x \)
• \( v'(x) = \frac{d}{dx}(x^2 + 3) = 2x \)

The quotient rule for derivatives states: \[ y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \] Substitute the derivatives and the original functions: \[ y' = \frac{14x(x^2 + 3) - 2x(7x^2 + 2)}{(x^2 + 3)^2} \]

Simplify the expression for \( y' \): \[ y' = \frac{14x^3 + 42x - 14x^3 - 4x}{(x^2 + 3)^2} \] \[ y' = \frac{38x}{(x^2 + 3)^2} \]

Final Answer