Tangent Equation Cd2315

1. **State the problem:** We need to find the equation of the tangent line to the curve $$y = 2^x + 2^{-x}$$ at the point $$\left(2, \frac{17}{4}\right)$$. 2. **Recall the formula for the tangent line:** The equation of the tangent line at a point $$x = a$$ is given by: $$ y = f(a) + f'(a)(x - a) $$ where $$f'(a)$$ is the derivative of $$f(x)$$ evaluated at $$x = a$$. 3. **Find the derivative $$f'(x)$$:** Given $$f(x) = 2^x + 2^{-x}$$, using the derivative rule $$\frac{d}{dx} a^x = a^x \ln(a)$$, we get: $$ f'(x) = 2^x \ln(2) - 2^{-x} \ln(2) = \ln(2)(2^x - 2^{-x}) $$ 4. **Evaluate $$f'(x)$$ at $$x=2$$:** $$ f'(2) = \ln(2) \left(2^2 - 2^{-2}\right) = \ln(2) \left(4 - \frac{1}{4}\right) = \ln(2) \times \frac{15}{4} $$ 5. **Write the tangent line equation:** Using $$a=2$$, $$f(2) = \frac{17}{4}$$, and $$f'(2) = \ln(2) \times \frac{15}{4}$$, $$ y = \frac{17}{4} + \left(\ln(2) \times \frac{15}{4}\right)(x - 2) $$ **Final answer:** $$ y = \frac{17}{4} + \frac{15}{4} \ln(2) (x - 2) $$