Chain Rule

1 Find the equation of the tangent to the curve \(y = \sqrt { 4 x + 1 }\) at the point ( 2,3 ).

1. The curve \(C\) has equation

$$y = ( 3 x - 2 ) ^ { 6 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) Given that the point \(P \left( \frac { 1 } { 3 } , 1 \right)\) lies on \(C\),
2. find the equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.

9 The point \(A ( 2,2 )\) lies on the curve \(y = x ^ { 2 } - 2 x + 2\).

1. Find the equation of the tangent to the curve at \(A\).
   The normal to the curve at \(A\) intersects the curve again at \(B\).
2. Find the coordinates of \(B\).
   The tangents at \(A\) and \(B\) intersect each other at \(C\).
3. Find the coordinates of \(C\).

5 A curve has equation \(y = 8 x + ( 2 x - 1 ) ^ { - 1 }\). Find the values of \(x\) at which the curve has a stationary point and determine the nature of each stationary point, justifying your answers.

10 The equation of a curve is \(y = 2 x + \frac { 8 } { x ^ { 2 } }\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point on the curve and determine the nature of the stationary point.
3. Show that the normal to the curve at the point \(( - 2 , - 2 )\) intersects the \(x\)-axis at the point \(( - 10,0 )\).
4. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).

2 The surface \(S\) has equation \(z = x ^ { 2 } y - 8 x y ^ { 2 } + \frac { x } { y }\) for \(y \neq 0\).

1. (a) Find the following.
   • \(\frac { \partial z } { \partial x }\)
   • \(\frac { \partial z } { \partial y }\)
     (b) Find the coordinates of all stationary points of \(S\).
   • Find all four second partial derivatives of \(z\) with respect to \(x\) and/or \(y\).

8 A curve \(y = \mathrm { f } ( x )\) has a stationary point at \(( 3,7 )\) and is such that \(\mathrm { f } ^ { \prime \prime } ( x ) = 36 x ^ { - 3 }\).

1. State, with a reason, whether this stationary point is a maximum or a minimum.
2. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ( x )\).

9 The function f is defined for \(x > 0\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - \frac { 2 } { x ^ { 2 } }\). The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( 2,6 )\).

1. Find the equation of the normal to the curve at \(P\).
2. Find the equation of the curve.
3. Find the \(x\)-coordinate of the stationary point and state with a reason whether this point is a maximum or a minimum.

3．Given that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( u \sqrt { } x ) = \frac { \mathrm { d } u } { \mathrm {~d} x } \times \frac { \mathrm { d } ( \sqrt { } x ) } { \mathrm { d } x } , \quad 0 < x < \frac { 1 } { 2 }$$ where \(u\) is a function of \(x\) ，and that \(u = 4\) when \(x = \frac { 3 } { 8 }\) ，find \(u\) in terms of \(x\) ．
（9）

1 Differentiate \(x + \sqrt { x ^ { 3 } }\).

4 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when

1. \(y = 2 x ^ { - 5 }\),
2. \(y = \sqrt [ 3 ] { x }\).

2. (i) Differentiate with respect to \(x\) $$2 x ^ { 3 } + \sqrt { } x + \frac { x ^ { 2 } + 2 x } { x ^ { 2 } }$$ (ii) Evaluate $$\int _ { 1 } ^ { 4 } \left( \frac { x } { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$

6 The variables \(x , y\) and \(z\) can take only positive values and are such that $$z = 3 x + 2 y \quad \text { and } \quad x y = 600 .$$

1. Show that \(z = 3 x + \frac { 1200 } { x }\).
2. Find the stationary value of \(z\) and determine its nature.

4 Variables \(u , x\) and \(y\) are such that \(u = 2 x ( y - x )\) and \(x + 3 y = 12\). Express \(u\) in terms of \(x\) and hence find the stationary value of \(u\).

1. Prove the identity \(\frac { \sin \theta - \cos \theta } { \sin \theta + \cos \theta } \equiv \frac { \tan \theta - 1 } { \tan \theta + 1 }\).
2. Hence solve the equation \(\frac { \sin \theta - \cos \theta } { \sin \theta + \cos \theta } = \frac { \tan \theta } { 6 }\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8 A hollow circular cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is \(192 \pi \mathrm {~cm} ^ { 2 }\). The cylinder has a radius of \(r \mathrm {~cm}\) and a height of \(h \mathrm {~cm}\).

1. Express \(h\) in terms of \(r\) and show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = \frac { 1 } { 2 } \pi \left( 192 r - r ^ { 3 } \right) .$$ Given that \(r\) can vary,
2. find the value of \(r\) for which \(V\) has a stationary value,
3. find this stationary value and determine whether it is a maximum or a minimum.

4 Find the second derivative of \(\left( x ^ { 2 } + 5 \right) ^ { 4 }\), giving your answer in factorised form.

1. $$y = 4 x ^ { 3 } - 7 x ^ { 2 } + 5 x - 10$$

1. Find in simplest form
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Hence find the exact value of \(x\) when \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\)

4. \(y = \arctan ( \sqrt { } x ) , \quad x > 0,0 < y < \frac { \pi } { 2 }\).

1. Find the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(\mathrm { x } = \frac { 1 } { 4 }\).
2. Show that \(2 x ( 1 + x ) \frac { d ^ { 2 } y } { d x ^ { 2 } } + ( 1 + 3 x ) \frac { d y } { d x } = 0\).

2 An increasing function, f , is defined for \(x > n\), where \(n\) is an integer. It is given that \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 6 x + 8\). Find the least possible value of \(n\).

9 The equation of a curve is \(y = 4 + 5 x + 6 x ^ { 2 } - 3 x ^ { 3 }\).

1. Find the set of values of \(x\) for which \(y\) decreases as \(x\) increases.
2. It is given that \(y = 9 x + k\) is a tangent to the curve. Find the value of the constant \(k\).

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 } ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 2 x\) for \(\frac { 1 } { 2 } < x < a\). It is given that f is a decreasing function. Find the maximum possible value of the constant \(a\).

6 The equation of a curve is \(y = 2 + \sqrt { 25 - x ^ { 2 } }\).
Find the coordinates of the point on the curve at which the gradient is \(\frac { 4 } { 3 }\).

11. A curve \(C\) has equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 5 x + 2\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2 . The \(x\)-coordinate of \(P\) is 3 .
2. Find the \(x\)-coordinate of \(Q\).
3. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. This tangent intersects the coordinate axes at the points \(R\) and \(S\).
4. Find the length of \(R S\), giving your answer as a surd.

6 The equation of a curve is \(y = 2 + \sqrt { 25 - x ^ { 2 } }\).
Find the coordinates of the point on the curve at which the gradient is \(\frac { 4 } { 3 }\).

8 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + a x + b\). The curve has stationary points at \(( - 1,2 )\) and \(( 3 , k )\). Find the values of the constants \(a , b\) and \(k\).

3 A curve has equation \(y = a x ^ { \frac { 1 } { 2 } } - 2 x\), where \(x > 0\) and \(a\) is a constant. The curve has a stationary point at the point \(P\), which has \(x\)-coordinate 9 . Find the \(y\)-coordinate of \(P\).

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that $$\mathrm { f } ^ { \prime } ( x ) = \left( \frac { 1 } { 2 } x + k \right) ^ { - 2 } - ( 1 + k ) ^ { - 2 }$$ where \(k\) is a constant. The curve has a minimum point at \(x = 2\).

1. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) in terms of \(k\) and \(x\), and hence find the set of possible values of \(k\).
   It is now given that \(k = - 3\) and the minimum point is at \(\left( 2,3 \frac { 1 } { 2 } \right)\).
2. Find \(\mathrm { f } ( x )\).
3. Find the coordinates of the other stationary point and determine its nature.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 Differentiate \(\sqrt [ 3 ] { 1 + 6 x ^ { 2 } }\).

4 For each of the following curves, find the exact gradient at the point indicated:

1. \(y = 3 \cos 2 x - 5 \sin x\) at \(\left( \frac { 1 } { 6 } \pi , - 1 \right)\),
2. \(x ^ { 3 } + 6 x y + y ^ { 3 } = 21\) at \(( 1,2 )\).

1 Given that \(y = ( 1 + 6 x ) ^ { \frac { 1 } { 3 } }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { y ^ { 2 } }\).

4

1. Given that \(x = ( 4 t + 9 ) ^ { \frac { 1 } { 2 } }\) and \(y = 6 \mathrm { e } ^ { \frac { 1 } { 2 } x + 1 }\), find expressions for \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) when \(t = 4\), giving your answer correct to 3 significant figures.

1. Given that \(y = \arctan \left( \frac { 2 x } { 3 } \right)\),
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in its simplest form.
   2. Use integration by parts to find
   $$\int \arctan \left( \frac { 2 x } { 3 } \right) \mathrm { d } x$$

2 Given that \(y = \left( x ^ { 2 } + 5 \right) ^ { 12 }\),

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find \(\int 48 x \left( x ^ { 2 } + 5 \right) ^ { 11 } \mathrm {~d} x\).

8

1. Given that \(y = \sqrt [ 3 ] { 1 + 3 x ^ { 2 } }\), use the chain rule to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
2. Given that \(y ^ { 3 } = 1 + 3 x ^ { 2 }\), use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Show that this result is equivalent to the result in part (i).

Differentiate with respect to \(x\)

1. \(\quad \ln \left( x ^ { 2 } + 3 x + 5 \right)\)
2. \(\frac { \cos x } { x ^ { 2 } }\)

11
\includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-18_497_1049_264_548} The diagram shows the curve with equation \(y = 9 \left( x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } } \right)\). The curve crosses the \(x\)-axis at the point \(A\).

1. Find the \(x\)-coordinate of \(A\).
2. Find the equation of the tangent to the curve at \(A\).
3. Find the \(x\)-coordinate of the maximum point of the curve.
4. Find the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 9\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9
\includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-4_611_895_255_625} The diagram shows the curve \(y = \sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\).

1. By first differentiating \(\frac { 1 - x } { 1 + x }\), obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). Hence show that the gradient of the normal to the curve at the point \(( x , y )\) is \(( 1 + x ) \sqrt { } \left( 1 - x ^ { 2 } \right)\).
2. The gradient of the normal to the curve has its maximum value at the point \(P\) shown in the diagram. Find, by differentiation, the \(x\)-coordinate of \(P\).