Problem Set Problem 10 (1 point) Differentiate R(t)=(4t+e^t)(3-sqrt (t)) Answer: R'(t)=

To differentiate the function \( R(t) = (4t + e^t)(3 - \sqrt{t}) \), we will use the product rule. The product rule states that if you have a function \( h(t) = f(t)g(t) \), then its derivative is given by:<br /><br />\[ h'(t) = f'(t)g(t) + f(t)g'(t) \]<br /><br />Here, let:<br />\[ f(t) = 4t + e^t \]<br />\[ g(t) = 3 - \sqrt{t} \]<br /><br />First, we find the derivatives of \( f(t) \) and \( g(t) \):<br /><br />\[ f'(t) = 4 + e^t \]<br />\[ g'(t) = -\frac{1}{2\sqrt{t}} \]<br /><br />Now, apply the product rule:<br /><br />\[ R'(t) = f'(t)g(t) + f(t)g'(t) \]<br /><br />Substitute \( f(t) \), \( f'(t) \), \( g(t) \), and \( g'(t) \):<br /><br />\[ R'(t) = (4 + e^t)(3 - \sqrt{t}) + (4t + e^t)\left(-\frac{1}{2\sqrt{t}}\right) \]<br /><br />Simplify each term:<br /><br />\[ R'(t) = (4 + e^t)(3 - \sqrt{t}) - \frac{4t + e^t}{2\sqrt{t}} \]<br /><br />So, the derivative of \( R(t) \) is:<br /><br />\[ R'(t) = (4 + e^t)(3 - \sqrt{t}) - \frac{4t + e^t}{2\sqrt{t}} \]