Find curve equation from derivative (reverse chain rule / composite functions)

9 A curve which passes through \(( 0,3 )\) has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 2 } { ( x - 1 ) ^ { 3 } }\).

1. Find the equation of the curve.
   The tangent to the curve at \(( 0,3 )\) intersects the curve again at one other point, \(P\).
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(( 2 x + 1 ) ( x - 1 ) ^ { 2 } - 1 = 0\).
3. Verify that \(x = \frac { 3 } { 2 }\) satisfies this equation and hence find the \(y\)-coordinate of \(P\).

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { ( 3 x - 2 ) ^ { 3 } }\) and \(A ( 1 , - 3 )\) lies on the curve. A point is moving along the curve and at \(A\) the \(y\)-coordinate of the point is increasing at 3 units per second.

1. Find the rate of increase at \(A\) of the \(x\)-coordinate of the point.
2. Find the equation of the curve.

2 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { ( x - 3 ) ^ { 2 } } + x\). It is given that the curve passes through the point (2, 7). Find the equation of the curve.

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { ( x + 2 ) ^ { 2 } }\) for \(x > - 2\).

1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
2. The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x )\). It is given that the point \(( - 1 , - 1 )\) lies on the curve. Find the equation of the curve.

7 The curve \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = \frac { - 3 } { ( x + 2 ) ^ { 4 } }\).

1. The tangent at a point on the curve where \(x = a\) has gradient \(- \frac { 16 } { 27 }\). Find the possible values of \(a\).
2. Find \(\mathrm { f } ( x )\) given that the curve passes through the point \(( - 1,5 )\).

2 A curve has equation \(y = f ( x )\). It is given that \(f ^ { \prime } ( x ) = \frac { 1 } { \sqrt { } ( x + 6 ) } + \frac { 6 } { x ^ { 2 } }\) and that \(f ( 3 ) = 1\). Find \(f ( x )\).

A curve with equation \(y = \mathrm{f}(x)\) passes through the point \(A(3, 1)\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm{f}'(x) = (3x - 1)^{-\frac{1}{3}}\). Find the \(y\)-coordinate of \(B\). [6]