Find the Derivative of \( y=e^{x^{2}+2 x} \) a) \[ \prime=2(x+1) e^{x^{2}+2 x} \] b) 1 c) 2 d) 3

To find the derivative of \( y = e^{x^2 + 2x} \), we can use the chain rule. The outer function is \( e^u \) where \( u = x^2 + 2x \). Differentiating, we have: 1. The derivative of \( e^u \) is \( e^u \). 2. Now, we differentiate \( u = x^2 + 2x \), which gives \( u' = 2x + 2 \). Applying the chain rule: \[ \frac{dy}{dx} = e^{x^2 + 2x} \cdot (2x + 2) = 2(x + 1)e^{x^2 + 2x} \] So, the correct answer is: a) \[ \prime=2(x+1) e^{x^{2}+2 x} \] --- Did you know that exponential functions, like \( e^x \), are unique because their rate of growth is proportional to their value at all points? This property makes them incredibly useful in modeling natural phenomena, such as population growth or radioactive decay! Also, if you ever feel stuck with derivatives, remember the power of practice! Graphing functions and their derivatives can improve your intuition. Mistakes often occur when forgetting to apply the chain rule properly; drawing out each step helps avoid confusion!