Questions: Find the derivative of each function. Simplify your answers where possible. a. g(x) = 1/2 x^10 + 8 sqrt(x) + 3x - 7 b. h(x) = 2 e^x - x^e + 2ex c. y = (2 x^2 - 1) / (x^2 + 4) d. G(x) = (4 x^2 - 1)(2 x^3 + 1) e. F(x) = e^x / (x + 1)

Solution Steps

To find the derivative of the function \( g(x) = \frac{1}{2} x^{10} + 8 \sqrt{x} + 3 x - 7 \), we apply the power rule to each term. The derivative is given by: \[ g'(x) = 5.0 x^{9} + 3 + \frac{4}{\sqrt{x}} \]

Next, we find the derivative of \( h(x) = 2 e^{x} - x^{e} + 2 e x \). Using the rules for derivatives of exponential functions and the power rule, we obtain: \[ h'(x) = \frac{2 x (e^{x} + e) - e x^{e}}{x} \]

Finally, we compute the derivative of \( y = \frac{2 x^{2} - 1}{x^{2} + 4} \) using the quotient rule. The derivative simplifies to: \[ y'(x) = \frac{18 x}{(x^{2} + 4)^{2}} \]

Final Answer