10. Demonstrate and explain how to find the derivative of the following functions. Be sure to explicitly denote which derivative rules (scalar multiple, sum/difference, quotient, etc.) you are using in your work. \[ f(w)=-2 e^{w}+3 \sin (w) \]

To find the derivative of the function \[ f(w) = -2 e^{w} + 3 \sin(w), \] we will apply the following derivative rules: 1. **Scalar Multiple Rule**: If \( c \) is a constant and \( f(w) \) is a differentiable function, then the derivative of \( c \cdot f(w) \) is \( c \cdot f'(w) \). 2. **Sum/Difference Rule**: The derivative of a sum (or difference) of functions is the sum (or difference) of their derivatives. 3. **Derivative of Exponential Function**: The derivative of \( e^{w} \) is \( e^{w} \). 4. **Derivative of Sine Function**: The derivative of \( \sin(w) \) is \( \cos(w) \). Now, let's differentiate \( f(w) \) step by step: 1. **Differentiate the first term**: - The first term is \( -2 e^{w} \). - Using the Scalar Multiple Rule, we have: \[ \frac{d}{dw}(-2 e^{w}) = -2 \cdot \frac{d}{dw}(e^{w}) = -2 \cdot e^{w}. \] 2. **Differentiate the second term**: - The second term is \( 3 \sin(w) \). - Again, using the Scalar Multiple Rule: \[ \frac{d}{dw}(3 \sin(w)) = 3 \cdot \frac{d}{dw}(\sin(w)) = 3 \cdot \cos(w). \] 3. **Combine the derivatives**: - Now, we apply the Sum/Difference Rule: \[ f'(w) = \frac{d}{dw}(-2 e^{w}) + \frac{d}{dw}(3 \sin(w)) = -2 e^{w} + 3 \cos(w). \] Thus, the derivative of the function \( f(w) \) is \[ f'(w) = -2 e^{w} + 3 \cos(w). \] This completes the differentiation process, clearly showing the rules applied at each step.