Find the equation of the tangent to the curve y = 2 ln(2e - x) at the point on the curve where x = e.
The first step is to find the y coordinate at the point on the curve where x = e. To do this we subsitute x = e into the equation of the curve, so y = 2ln(2e - x) becomes y = 2ln(2e - e). Simplifying this, we get y = 2ln(e). Using our knowledge of the natural logarithm, ln(e) = 1, so therefore y = 2.
Next, we find the gradient of the curve at the point x = e. We do this by differentiating the equation, and then subsituting the value for x in. Note that we have to use the chain rule.
dy/dx = -2/(2e - x) = -2/(2e - e) = -2/e.
Using all the information we've found, we can now produce the equation of the tangent line using the equation of a line formula, y - y0 = m(x - x0), where m is the gradient.
y - 2 = -2/e (x - e). Rearranging this, we get y = 4 - 2x/e as the equation of the tangent line.
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