Find dy/dx at a point

5 Find the gradient at the point \(( 0 , \ln 2 )\) on the curve with equation \(\mathrm { e } ^ { 2 y } = 5 - \mathrm { e } ^ { - x }\).

1 Fig. 7 shows the curve \(y = _ { x - 1 }\). It has a minimum at the point P . The line \(l\) is an asymptote to the curve. \begin{figure}[h] \end{figure}

1. Write down the equation of the asymptote \(l\).
2. Find the coordinates of P .
3. Using the substitution \(u = x - 1\), show that the area of the region enclosed by the \(x\)-axis, the curve and the lines \(x = 2\) and \(x = 3\) is given by $$\int _ { 1 } ^ { 2 } \left( u + 2 + \frac { 4 } { u } \right) \mathrm { d } u$$ Evaluate this area exactly.
4. Another curve is defined by the equation \(\mathrm { e } ^ { y } = \frac { x ^ { 2 } + 3 } { x - 1 }\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\) by differentiating implicitly. Hence find the gradient of this curve at the point where \(x = 2\).

2 Fig. 7 shows the curve defined implicitly by the equation $$y ^ { 2 } + y = x ^ { 9 } + 2 x$$ together with the line \(x = 2\). \begin{figure}[h] \end{figure} Not to scale Find the coordinates of the points of intersection of the line and the curve.
Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at each of these two points.

3 Find the gradient at the point \(( 0 , \ln 2 )\) on the curve with equation \(\mathrm { e } ^ { 2 y } = 5 - \mathrm { e } ^ { - x }\).

8

1. Find the gradient of the curve \(x ^ { 2 } + x y + y ^ { 2 } = 3\) at the point \(( - 1 , - 1 )\).
2. A curve \(C\) has parametric equations $$x = 2 t ^ { 2 } - 1 , y = t ^ { 3 } + t$$
   1. Find the coordinates of the point on \(C\) at which the tangent is parallel to the \(y\)-axis.
   2. Find the values of \(t\) for which \(x\) and \(y\) have the same rate of change with respect to \(t\).

7 A curve has equation \(( x + y ) ^ { 2 } = x y ^ { 2 }\). Find the gradient of the curve at the point where \(x = 1\).

15 You must show detailed reasoning in this question. The equation of a curve is $$y ^ { 3 } - x y + 4 \sqrt { x } = 4 .$$ Find the gradient of the curve at each of the points where \(y = 1\).

7 A curve has equation \(y ^ { 3 } + 2 \mathrm { e } ^ { - 3 x } y - x = k\), where \(k\) is a constant.
The point \(P \left( \ln 2 , \frac { 1 } { 2 } \right)\) lies on this curve.

1. Show that the exact value of \(k\) is \(q - \ln 2\), where \(q\) is a rational number.
2. Find the gradient of the curve at \(P\).

5 The curve \(C\) has equation $$3 x ^ { 2 } - 5 x y + \mathrm { e } ^ { 2 y - 4 } + 6 = 0$$ The point \(P\) with coordinates \(( 1,2 )\) lies on \(C\). The tangent to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\) and the normal to \(C\) at \(P\) meets the \(y\)-axis at the point \(B\). Find the exact area of triangle \(A B P\).

6 A curve has equation \(3 x y - 2 y ^ { 2 } = 4\).
Find the gradient of the curve at the point \(( 2,1 )\).

6 A curve is defined by the equation \(x ^ { 2 } y + y ^ { 3 } = 2 x + 1\).

1. Find the gradient of the curve at the point \(( 2,1 )\).
2. Show that the \(x\)-coordinate of any stationary point on this curve satisfies the equation $$\frac { 1 } { x ^ { 3 } } = x + 1$$ (4 marks)

5 A curve is defined by the equation $$x ^ { 2 } + x y = \mathrm { e } ^ { y }$$ Find the gradient at the point \(( - 1,0 )\) on this curve.

5 The point \(P ( 1 , a )\), where \(a > 0\), lies on the curve \(y + 4 x = 5 x ^ { 2 } y ^ { 2 }\).

1. Show that \(a = 1\).
2. Find the gradient of the curve at \(P\).
3. Find an equation of the tangent to the curve at \(P\).

5 A curve is defined by the equation \(4 x ^ { 2 } + y ^ { 2 } = 4 + 3 x y\).
Find the gradient at the point ( 1,3 ) on this curve.

12 A curve \(C\) has equation $$x ^ { 3 } \sin y + \cos y = A x$$ where \(A\) is a constant. \(C\) passes through the point \(P \left( \sqrt { 3 } , \frac { \pi } { 6 } \right)\) 12

1. Show that \(A = 2\) 12
2. (i) Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - 3 x ^ { 2 } \sin y } { x ^ { 3 } \cos y - \sin y }\) 12 (b) (ii) Hence, find the gradient of the curve at \(P\).
   12 (b) (iii) The tangent to \(C\) at \(P\) intersects the \(x\)-axis at \(Q\).
   Find the exact \(x\)-coordinate of \(Q\).

Fig. 7 shows the curve defined implicitly by the equation $$y^2 + y = x^3 + 2x,$$ together with the line \(x = 2\). \includegraphics{figure_7} Find the coordinates of the points of intersection of the line and the curve. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at each of these two points.

A curve is defined by the equation \(\sin 2x + \cos y = \sqrt{3}\).

1. Verify that the point P \((\frac{\pi}{6}, \frac{\pi}{6})\) lies on the curve. [1]
2. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point P. [5]

Fig. 6 shows part of the curve \(\sin 2y = x - 1\). P is the point with coordinates \((1.5, \frac{1}{12}\pi)\) on the curve. \includegraphics{figure_6}

1. Find \(\frac{dy}{dx}\) in terms of \(y\). Hence find the exact gradient of the curve \(\sin 2y = x - 1\) at the point P. [4]

The part of the curve shown is the image of the curve \(y = \arcsin x\) under a sequence of two geometrical transformations.

1. Find \(y\) in terms of \(x\) for the curve \(\sin 2y = x - 1\). Hence describe fully the sequence of transformations. [4]

A curve has equation \(x^3 y + \cos(\pi y) = 7\).

1. Find the exact value of the \(x\)-coordinate at the point on the curve where \(y = 1\). [2 marks]
2. Find the gradient of the curve at the point where \(y = 1\). [5 marks]

Find the gradient of the curve \(4x^2 + 2xy + y^2 = 12\) at the point \((1, 2)\). [4]