Find tangent equation at point

4 A curve has equation $$3 x ^ { 2 } - y ^ { 2 } - 4 \ln ( 2 y + 3 ) = 26$$ Find the equation of the tangent to the curve at the point \(( 3 , - 1 )\).

6 The equation of a curve is $$x ^ { 2 } y + y ^ { 2 } = 6 x$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 - 2 x y } { x ^ { 2 } + 2 y }\).
2. Find the equation of the tangent to the curve at the point with coordinates ( 1,2 ), giving your answer in the form \(a x + b y + c = 0\).

5 A curve has equation \(x ^ { 2 } + 2 y ^ { 2 } + 5 x + 6 y = 10\). Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
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5 A curve has equation $$y ^ { 3 } \sin 2 x + 4 y = 8$$ Find the equation of the tangent to the curve at the point where it crosses the \(y\)-axis.

6 The equation of a curve is $$x \ln y = 2 x + 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { y } { x ^ { 2 } }\).
2. Find the equation of the tangent to the curve at the point where \(y = 1\), giving your answer in the form \(a x + b y + c = 0\).

4 The equation of a curve is $$\sqrt { } x + \sqrt { } y = \sqrt { } a$$ where \(a\) is a positive constant.

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. The straight line with equation \(y = x\) intersects the curve at the point \(P\). Find the equation of the tangent to the curve at \(P\).

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find the equation of the tangent to the curve at the point \(( 2,1 )\), giving your answer in the form \(a x + b y + c = 0\).

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

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 y - x ^ { 2 } } { y ^ { 2 } - 3 x }\).
2. Find the equation of the tangent to the curve at the point ( 2,4 ), giving your answer in the form \(a x + b y = c\).

7 The equation of a curve is \(x ^ { 2 } + 4 x y + 2 y ^ { 2 } = 7\).

1. Find the equation of the tangent to the curve at the point \(( - 1,3 )\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
2. Show that there is no point on the curve at which the gradient is \(\frac { 1 } { 2 }\).

1. The point \(P\) lies on the curve with equation

$$x = ( 4 y - \sin 2 y ) ^ { 2 }$$ Given that \(P\) has \(( x , y )\) coordinates \(\left( p , \frac { \pi } { 2 } \right)\), where \(p\) is a constant,

1. find the exact value of \(p\) The tangent to the curve at \(P\) cuts the \(y\)-axis at the point \(A\).
2. Use calculus to find the coordinates of \(A\).

1. (a) Prove, by using logarithms, that

$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( 2 ^ { x } \right) = 2 ^ { x } \ln 2$$ The curve \(C\) has the equation $$2 x + 3 y ^ { 2 } + 3 x ^ { 2 } y + 12 = 4 \times 2 ^ { x }$$ The point \(P\), with coordinates \(( 2,0 )\), lies on \(C\).
(b) Find an equation of the tangent to \(C\) at \(P\).

1. Find an equation of the tangent to the curve

$$x ^ { 3 } + 3 x ^ { 2 } y + y ^ { 3 } = 37$$ at the point \(( 1,3 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
(6)

2. A curve \(C\) has the equation $$x ^ { 3 } - 3 x y - x + y ^ { 3 } - 11 = 0$$ Find an equation of the tangent to \(C\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. A curve \(C\) has equation

$$3 x ^ { 2 } + 2 x y - 2 y ^ { 2 } + 4 = 0$$ Find an equation for the tangent to \(C\) at the point ( 2,4 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
(6)

4. Find the equation of the tangent to the curve \(x = \cos ( 2 y + \pi )\) at \(\left( 0 , \frac { \pi } { 4 } \right)\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be found.

3. The curve \(C\) has equation \(x = 8 y \tan 2 y\) The point \(P\) has coordinates \(\left( \pi , \frac { \pi } { 8 } \right)\)

1. Verify that \(P\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at \(P\) in the form \(a y = x + b\), where the constants \(a\) and \(b\) are to be found in terms of \(\pi\).

5. The point \(P\) lies on the curve with equation $$x = ( 4 y - \sin 2 y ) ^ { 2 }$$ Given that \(P\) has \(( x , y )\) coordinates \(\left( p , \frac { \pi } { 2 } \right)\), where \(p\) is a constant,

1. find the exact value of \(p\). The tangent to the curve at \(P\) cuts the \(y\)-axis at the point \(A\).
2. Use calculus to find the coordinates of \(A\).

1. The curve \(C\) has equation

$$x y ^ { 2 } = x ^ { 2 } y + 6 \quad x \neq 0 \quad y \neq 0$$ Find an equation for the tangent to \(C\) at the point \(P ( 2,3 )\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
(6)

1. A curve \(C\) is described by the equation

$$3 x ^ { 2 } + 4 y ^ { 2 } - 2 x + 6 x y - 5 = 0$$ Find an equation of the tangent to \(C\) at the point \(( 1 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. The curve \(C\) has the equation

$$\cos 2 x + \cos 3 y = 1 , \quad - \frac { \pi } { 4 } \leqslant x \leqslant \frac { \pi } { 4 } , \quad 0 \leqslant y \leqslant \frac { \pi } { 6 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) lies on \(C\) where \(x = \frac { \pi } { 6 }\).
2. Find the value of \(y\) at \(P\).
3. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c \pi = 0\), where \(a , b\) and \(c\) are integers. \section*{LU}

2. A curve \(C\) has the equation $$x ^ { 3 } + 2 x y - x - y ^ { 3 } - 20 = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find an equation of the tangent to \(C\) at the point \(( 3 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

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

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { 2 y ^ { 2 } - 3 }$$
2. Find an equation for the tangent to the curve at the point where \(y = \frac { 1 } { 2 }\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

1. A curve has the equation

$$x ^ { 2 } - 3 x y - y ^ { 2 } = 12$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find an equation for the tangent to the curve at the point \(( 2 , - 2 )\).

1. A curve has the equation

$$4 \cos x + 2 \sin y = 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y\).
2. Find an equation for the tangent to the curve at the point ( \(\frac { \pi } { 3 } , \frac { \pi } { 6 }\) ), giving your answer in the form \(a x + b y = c\), where \(a\) and \(b\) are integers.