Questions: Use the product rule to find the derivative. y=(4x^2+3)(2x-3) y'=

Transcript text: Use the product rule to find the derivative. \[ \begin{array}{l} y=\left(4 x^{2}+3\right)(2 x-3) \\ y^{\prime}=\square \end{array} \] $\square$

Solution

Solution Steps

Step 1: Identify the functions

Given that $y = u(x)v(x)$, where $u(x) = 4_x^2 + 3$ and $v(x) = 2_x - 3$.

Step 2: Differentiate $u(x)$ and $v(x)$ with respect to $x$

The derivative of $u(x)$, $u'(x)$, is $ 8 x $. The derivative of $v(x)$, $v'(x)$, is $ 2 $.

Step 3: Apply the Product Rule

The Product Rule states that if $y = u(x)v(x)$, then $y' = u'(x)v(x) + u(x)v'(x)$. Substituting the derivatives, we get $y' = 8 x_2_x - 3 + 4_x^2 + 3_2$.

Step 4: Simplify the expression

After simplification, $y' = 24 x^{2} - 24 x + 6$.