Questions: Compute derivatives, P'(t), for the following functions. Simplify your answer. (a) P(t)=6 e^(6 t)+3 t^4-ln(2 t) (b) P(t)=(2 t^12-5 e^t-6)^(9 / 5)

Solution Steps

To solve these problems, we need to compute the derivatives of the given functions with respect to \( t \). For part (a), we will apply the rules of differentiation for exponential, polynomial, and logarithmic functions. For part (b), we will use the chain rule to differentiate the composite function.

Given the function \[ P(t) = 6 e^{6t} + 3t^4 - \ln(2t), \] we differentiate it with respect to \( t \). The derivative is calculated as follows: \[ P'(t) = \frac{d}{dt}(6 e^{6t}) + \frac{d}{dt}(3t^4) - \frac{d}{dt}(\ln(2t)). \] This results in: \[ P'(t) = 36 e^{6t} + 12t^3 - \frac{1}{t}. \] Thus, the simplified derivative for part (a) is: \[ P'(t) = 12t^3 + 36 e^{6t} - \frac{1}{t}. \]

For the function \[ P(t) = (2t^{12} - 5 e^{t} - 6)^{\frac{9}{5}}, \] we apply the chain rule to differentiate: \[ P'(t) = \frac{9}{5}(2t^{12} - 5 e^{t} - 6)^{\frac{4}{5}} \cdot \frac{d}{dt}(2t^{12} - 5 e^{t} - 6). \] Calculating the derivative of the inner function gives: \[ \frac{d}{dt}(2t^{12} - 5 e^{t} - 6) = 24t^{11} - 5 e^{t}. \] Thus, the derivative for part (b) is: \[ P'(t) = \frac{9}{5}(2t^{12} - 5 e^{t} - 6)^{\frac{4}{5}}(24t^{11} - 5 e^{t}). \]

Final Answer