Find dy/dx at a point

3 The equation of a curve is \(\cos 3 x + 5 \sin y = 3\).
Find the gradient of the curve at the point \(\left( \frac { 1 } { 9 } \pi , \frac { 1 } { 6 } \pi \right)\).

3 A curve has equation \(\mathrm { e } ^ { 2 x } \cos 2 y + \sin y = 1\).
Find the exact gradient of the curve at the point \(\left( 0 , \frac { 1 } { 6 } \pi \right)\).

2 A curve has equation \(x ^ { 2 } \ln y + y ^ { 2 } + 4 x = 9\).
Find the gradient of the curve at the point \(( 2,1 )\).

5 A curve has equation \(x ^ { 2 } + 4 x \cos 3 y = 6\).
Find the exact value of the gradient of the normal to the curve at the point \(\left( \sqrt { 2 } , \frac { 1 } { 12 } \pi \right)\).

5 A curve has equation \(4 \mathrm { e } ^ { 2 x } y + y ^ { 2 } = 21\).
Find the gradient of the curve at the point \(( 0 , - 7 )\).

3 A curve has equation \(6 \mathrm { e } ^ { - x } y ^ { 2 } + \mathrm { e } ^ { 2 x } - 12 y + 7 = 0\).
Find the gradient of the curve at the point \(( \ln 3,2 )\).

7

1. Find the gradient of the curve $$3 \ln x + 4 \ln y + 6 x y = 6$$ at the point \(( 1,1 )\).
2. The parametric equations of a curve are $$x = \frac { 10 } { t } - t , \quad y = \sqrt { } ( 2 t - 1 ) .$$ Find the gradient of the curve at the point \(( - 3,3 )\).

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

6 The equation of a curve is \(3 x ^ { 2 } - 4 x y + y ^ { 2 } = 45\).

1. Find the gradient of the curve at the point \(( 2 , - 3 )\).
2. Show that there are no points on the curve at which the gradient is 1 .

3 Find the gradient of the curve \(x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 1\) at the point with coordinates \(( 1,3 )\).

4 A curve has equation \(3 \mathrm { e } ^ { 2 x } y + \mathrm { e } ^ { x } y ^ { 3 } = 14\). Find the gradient of the curve at the point \(( 0,2 )\).

4 Find the gradient of the curve $$x ^ { 2 } \sin y + \cos 3 y = 4$$ at the point \(\left( 2 , \frac { 1 } { 2 } \pi \right)\).

4 Find the gradient of the curve $$4 x + 3 y \mathrm { e } ^ { 2 x } + y ^ { 2 } = 10$$ at the point \(( 0,2 )\).

8 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } + 2 x y + 8 = 0\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
   The tangent to the curve at the point where \(x = 0\) and the tangent at the point where \(y = 0\) intersect at the acute angle \(\alpha\).
2. Find the exact value of \(\tan \alpha\).

4 The equation of a curve is \(\mathrm { ye } ^ { 2 \mathrm { x } } + \mathrm { y } ^ { 2 } \mathrm { e } ^ { \mathrm { x } } = 6\).
Find the gradient of the curve at the point where \(y = 1\).

3 The equation of a curve is \(\ln ( x + y ) = 3 x ^ { 2 } y\).
Find the gradient of the curve at the point \(( 1,0 )\).

5 The variables \(x\) and \(y\) are such that \(y = 0\) when \(x = 0\) and $$( x + 1 ) y + ( x + y + 1 ) ^ { 3 } = 1$$

1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 3 } { 4 }\) when \(x = 0\).
2. Find the Maclaurin's series for \(y\) up to and including the term in \(x ^ { 2 }\).

3. A curve \(C\) has equation $$3 ^ { x } + 6 y = \frac { 3 } { 2 } x y ^ { 2 }$$ Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) with coordinates (2, 3). Give your answer in the form \(\frac { a + \ln b } { 8 }\), where \(a\) and \(b\) are integers.

1. A curve \(C\) has equation

$$3 ^ { x } + x y = x + y ^ { 2 } , \quad y > 1$$ The point \(P\) with coordinates \(( 4,11 )\) lies on \(C\).
Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). Give your answer in the form \(a + b \ln 3\), where \(a\) and \(b\) are rational numbers.

3. The curve \(C\) has equation $$3 y ^ { 2 } - 11 x ^ { 2 } + 11 x y = 20 y - 36 x + 28$$

1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P ( 4 , k )\), where \(k\) is a constant, lies on \(C\).
   Given that \(k < 0\)
2. find the value of the gradient of \(C\) at \(P\)

5. A curve is described by the equation $$x ^ { 3 } - 4 y ^ { 2 } = 12 x y$$

1. Find the coordinates of the two points on the curve where \(x = - 8\).
2. Find the gradient of the curve at each of these points.

A curve \(C\) has the equation \(y ^ { 2 } - 3 y = x ^ { 3 } + 8\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Hence find the gradient of \(C\) at the point where \(y = 3\).

3. A curve \(C\) has equation $$2 ^ { x } + y ^ { 2 } = 2 x y$$ Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) with coordinates \(( 3,2 )\).

1. Find the gradient of the curve with equation

$$\ln y = 2 x \ln x , \quad x > 0 , y > 0$$ at the point on the curve where \(x = 2\). Give your answer as an exact value.

7 Fig. 7 shows the curve defined implicitly by the equation $$y ^ { 2 } + y = x ^ { 3 } + 2 x ,$$ together with the line \(x = 2\). \begin{figure}[h] \end{figure} Fig. 7 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.