This was posted in the "Pre Calculus" section so Prove It used a method that did not require the derivative.
Since you do mention the derivative, no, the derivative is not 0 where the line y= x+ c is tangent to it. A line is tangent to a curve where its slope is the same as the derivative. y= x+ c has slope 1 so you are looking for a place where the derivative is 1, not 0.
Here is a method using the derivative.
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The derivative of the function is (power rule on sum of functions).
The line has a slope equal to . Thus, you are looking for the point on the curve of where the slope is equal to . So, you must solve for . Hmm, .
Say (to make it less confusing). You know that the tangent to the curve of in a point of absciss is equal to :
Substitute :
That is : . Therefore .