# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | Oct-Nov | (P2-9709/02) | Q#6

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Question

Find the exact coordinates of the point on the curve at which .

Solution

We are given that; We are required to find; Second derivative is the derivative of the derivative. If we have derivative of the curve as , then  expression for the second derivative of the curve is; Therefore first we find; If and are functions of , and if , then; If , then;  Rule for differentiation of is: Rule for differentiation natural exponential function , or ;      Next we find .  Rule for differentiation of is:    Rule for differentiation natural exponential function , or ;  If and are functions of , and if , then; If , then;  As demonstrated above; Therefore;   We are given that at the desired point; Hence;     Corresponding values of y coordinate can be found by substituting values of x in equation of the line  or the curve.

Therefore;   Hence, coordinates of the point on the curve where are .