Question

Multiple Choice Which of the following gives the slope of the tangent line to the graph of \(y=2^{1-x}\) at \(x=2 ?\) . E $$(\mathbf{A})-\frac{1}{2} \quad(\mathbf{B}) \frac{1}{2} \quad(\mathbf{C})-2 \quad\( (D) \)2 \quad(\mathbf{E})-\frac{\ln 2}{2}$$

Short answer

The slope of the tangent line to the graph of \(y=2^{1-x}\) at \(x=2\) is \(-\frac{\ln 2}{2}\), so the correct multiple choice answer is (E)

Step-by-step solution

Set up the function

Our function is \(y = 2^{1-x}\). We first rewrite it into a easier form for differentiation. We rewrite as \(y = 2^{1-x} = e^{(1-x)\ln 2}\)

Calculate the derivative

The derivative of this function can be found using the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Applying the rule, the derivative \(y' = \frac{d}{dx}\left[ e^{(1-x)\ln 2}\right]= - \ln 2 \cdot e^{(1-x)\ln 2}\)

Evaluate the derivative at \(x=2\)

Substitute \(x = 2\) into the derivative gives \(y'=-\ln 2 \cdot e^{(1-2)\ln 2}=-\ln 2 \cdot e^{-\ln 2}=-\ln 2 \cdot \frac{1}{2}=-\frac{\ln 2}{2}\)