Find an equation of the tangent line to the curve $y=e^x$ at the point $(1,e)$.

• A. $y=x+e$
• B. $y=e(x+1)$
• C. $y=e^x+1$
• D. $y=e(x-1)+e$

Answer: (D)

Verified step by step guidance

1. Step 1: Recall the formula for the equation of a tangent line to a curve at a given point. The equation is y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency and m is the slope of the tangent line.
2. Step 2: To find the slope of the tangent line, differentiate the given curve y = e^x with respect to x. The derivative of e^x is e^x, so the slope m at any point x is m = e^x.
3. Step 3: Evaluate the slope at the given point (1, e). Substituting x = 1 into m = e^x, we get m = e^1 = e.
4. Step 4: Substitute the point of tangency (x₁, y₁) = (1, e) and the slope m = e into the tangent line formula y - y₁ = m(x - x₁). This gives y - e = e(x - 1).
5. Step 5: Simplify the equation y - e = e(x - 1) to get the final equation of the tangent line: y = e(x - 1) + e.