Question

In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=e^{-x}$$

Short answer

The derivative of the function \(y = e^{-x}\) is \(\frac{dy}{dx} = -e^{-x}\)

Step-by-step solution

Identify the Function

The function given is \(y = e^{-x}\), an exponential function where the base is e (natural number) and the exponent is -x.

Apply the Chain Rule

To find the derivative of this function, we need to use the chain rule. The chain rule states that the derivative of a composition of functions is the product of the derivative of the inner function and the derivative of the outer function evaluated at the inner function. In our case, the outer function is \(e^x\) and the inner function is \(-x\). The derivative of \(e^x\) is still \(e^x\) and the derivative of \(-x\) is \(-1\). So, applying the chain rule, we get: \(\frac{dy}{dx} = e^{-x} * -1 \).

Simplify the Equation

By simplifying, we can write the derivative as: \(\frac{dy}{dx} = -e^{-x} \)