# How do you use the first and second derivatives to sketch #y = e^(1-2x)#?

See argument below.

Apply chain rule

Apply power rule

graph{e^(1-2x) [-5.69, 5.406, -1.722, 3.827]}

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To sketch ( y = e^{1-2x} ) using the first and second derivatives:

- Find the first derivative: ( \frac{dy}{dx} = -2e^{1-2x} ).
- Find critical points by setting the first derivative equal to zero and solving for ( x ).
- Find the second derivative: ( \frac{d^2y}{dx^2} = 4e^{1-2x} ).
- Determine the concavity of the function by analyzing the sign of the second derivative.
- Sketch the curve based on the behavior of the first and second derivatives, including critical points and concavity.

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