Questions: Quiz 2 Differentiate each of the following with respect to x. (a) y = πx^3 + 5/x - 6x^(1/3) dy/dx = 3πx^2 + 5x^(-1) - 6x^(-2/3) (b) (4x-7)^8 / 5

Transcript text: Quiz 2 Differentiate each of the following with respect to $x$. (a) \[ \begin{aligned} y & =\pi x^{3}+\frac{5}{x}-6 \sqrt[3]{x} \\ \frac{d y}{d x} & =3 \pi x^{2}+5 x^{-1}-6 x^{-\frac{2}{3}} \\ \end{aligned} \] (b) $\frac{(4 x-7)^{8}}{5}$

Solution

Solution Steps

Step 1: Differentiate $ y_a $

Given the function \[ y_a = \pi x^{3} + \frac{5}{x} - 6 \sqrt[3]{x} \] we apply the power rule and the constant multiple rule to differentiate each term. The derivative is calculated as follows: \[ \frac{d y_a}{d x} = 3 \pi x^{2} - 5 x^{-2} - 2 x^{-\frac{2}{3}} \]

Step 2: Differentiate $ y_b $

For the function \[ y_b = \frac{(4x - 7)^{8}}{5} \] we use the chain rule along with the power rule. The derivative is computed as: \[ \frac{d y_b}{d x} = \frac{8}{5} (4x - 7)^{7} \cdot 4 = \frac{32(4x - 7)^{7}}{5} \]