Integration after differentiation

A question is this type if and only if it asks students to first differentiate from first principles, then use the result to find the equation of a curve by integration given a point.

2 questions · Moderate -0.8

1.07g Differentiation from first principles: for small positive integer powers of x 1.08a Fundamental theorem of calculus: integration as reverse of differentiation

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OCR H240/01 2023 June Q3

7 marks Moderate -0.8

Given that \(\mathrm { f } ( x ) = x ^ { 2 } + 2 x\), use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + 2\).
The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x + 2\) and the curve passes through the point \(( - 1,5 )\). Find the equation of the curve.

OCR H240/01 2018 December Q5

8 marks Moderate -0.8

Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x\), use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - 4\).
Find the equation of the curve through \(( 2,7 )\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - 4\).