Questions: What is the derivative of the function f(x)=e^[(x+1)^2] ?

Solution Steps

To find the derivative of the function \( f(x) = e^{(x+1)^2} \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is \( f'(g(x)) \cdot g'(x) \). Here, let \( u = (x+1)^2 \), so \( f(x) = e^u \). We need to find \( \frac{d}{dx} e^u \) and \( \frac{d}{dx} (x+1)^2 \), and then multiply these derivatives together.

We start with the function \( f(x) = e^{(x+1)^2} \).

To find the derivative of \( f(x) \), we use the chain rule. Let \( u = (x+1)^2 \). Then \( f(x) = e^u \).

First, we find the derivative of the inner function \( u \) with respect to \( x \): \[ \frac{d}{dx} (x+1)^2 = 2(x+1) \]

Next, we find the derivative of the outer function \( e^u \) with respect to \( u \): \[ \frac{d}{du} e^u = e^u \]

Using the chain rule, we multiply the derivatives of the inner and outer functions: \[ \frac{d}{dx} f(x) = \frac{d}{du} e^u \cdot \frac{d}{dx} (x+1)^2 = e^{(x+1)^2} \cdot 2(x+1) \]

Final Answer