Questions: Problem 5. (1 point) Differentiate B(q) = (e^4) sin (q^3+2 q^2) Answer: B'(q) =

Solution Steps

To differentiate the function \( B(q) = \left(e^{4}\right) \sin \left(q^{3}+2 q^{2}\right) \), we will apply the chain rule and the constant multiple rule. The constant \( e^4 \) can be factored out, and we differentiate the sine function using the chain rule, which involves differentiating the inner function \( q^3 + 2q^2 \).

We start with the function \( B(q) = e^{4} \sin(q^{3} + 2q^{2}) \). To differentiate this function, we apply the product and chain rules. The constant \( e^{4} \) remains unchanged during differentiation.

Using the chain rule, we differentiate the sine function: \[ B'(q) = e^{4} \cdot \cos(q^{3} + 2q^{2}) \cdot \frac{d}{dq}(q^{3} + 2q^{2}) \] Next, we differentiate the inner function \( q^{3} + 2q^{2} \): \[ \frac{d}{dq}(q^{3} + 2q^{2}) = 3q^{2} + 4q \]

Now we can combine the results to express \( B'(q) \): \[ B'(q) = e^{4} \cdot \cos(q^{3} + 2q^{2}) \cdot (3q^{2} + 4q) \] Thus, the final expression for the derivative is: \[ B'(q) = (3q^{2} + 4q) e^{4} \cos(q^{3} + 2q^{2}) \]

Final Answer