Implicit Differentiation (Cambridge International AS Mathematics): Exam Questions

Find an expression for $\frac{\text{d}y}{\text{d}x}$ when

(i) $x^{2} + y = 3$,

(ii) $5 x^{4} + y^{2} - 4 = 0$,

(iii) $\text{sin} 3 x - 3 y = 0$,

(iv) $\text{e}^{x} + \text{e}^{y} = 2 x$.

The equation of a curve is $3 y^{2} - 2 x^{3} = 10$.

Find the gradient of the curve at the point $( 1 , 2 )$.

The equation of a curve is $x - \text{sin} y = 0$.

Show that $\frac{\text{d}y}{\text{d}x} = \text{sec} y$.

A curve has equation $y^{2} - 4 x + 2 = 0$.

Show that the curve intersects the $x$-axis at the point \left(\dfrac{1}{2}, \; 0\right).

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

Show that the curve has no stationary points.

Show that the point \left(-\dfrac{4}{\pi}, \; \dfrac{\pi}{2}\right) lies on the curve with equation $2 \text{cos} 2 y = x y$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

The equation of a curve is $12 x^{2} - 4 y^{2} + 24 = 0$.

Find the gradient of the curve at the point $( 1 , 3 )$.

Hence find the equation of the tangent to the curve at the point $( 1 , 3 )$.

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

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

Hence show that the stationary points of the curve lie on the line $y = 6 x$.

The equation of a curve is $x^{3} + 9 x y^{2} = 54$.

Find the gradient of the curve at the point $( 3 , 1 )$.

Hence find the equation of the normal to the curve at the point $( 3 , 1 )$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.

The equation of a curve is $3 x^{2} y + 4 x - y = 41$.

Find the gradient of the curve at the point $( 2 , 3 )$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$ for the following

(i) $2 x y + y^{2} = 4$

(ii) $3 \text{sin} y - y = 2 x - 1$

The equation of a curve is $\frac{1}{5} x^{2} \text{e}^{y} = 5$.

Show that the curve intersects the $x$-axis at the points $( - 5 , 0 )$ and $( 5 , 0 )$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

Hence find the gradient of the curve at each of these points.

The equation of a curve is $3 \text{tan} y = 2 x y$.

Show that the point $( 0 , π )$ lies on the curve.

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

Find the gradient of the curve at the point $( 0 , π )$.

Hence find the equation of the tangent to the curve at the point $( 0 , π )$.

The equation of a curve is $\text{ln} y = 1 - x y$. The point $P$ with coordinates $( 1 , 1 )$ lies on the curve.

Show that $\frac{\text{d}y}{\text{d}x} = - \frac{y^{2}}{1+xy}$.

Find the gradient of the curve at $P$, and hence find the gradient of the normal to the curve at $P$.

Hence find the equation of the normal to the curve at $P$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.

The equation of a curve is $2 x^{2} - y = x y^{2}$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

Show that $\frac{\text{d}y}{\text{d}x} = 0$ when $4 x = y^{2}$.

Hence find the coordinates of the stationary points of the curve.

The equation of a curve is $\text{e}^{xy} = y - x$.

Find the coordinates of the points where the curve meets the $x$-axis and the $y$-axis.

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

Show that the tangents to the curve at these two points have equations $y = 2 x + 1$ and $2 y = x + 1$.

The two tangents intersect at the point $Q$. Find the distance $O Q$, where $O$ is the origin.

Use implicit differentiation to show that

\dfrac{\text{d}}{\text{d}x}\left(a^{x}\right) = a^{x}\,\text{ln} \; a

where $a$ is a constant.

Show that

\dfrac{\text{d}}{\text{d}x}\left(a^{kx}\right) = k a^{kx}\,\text{ln} \; a

where $a$ and $k$ are constants.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$ for the following

(i) $2 y \text{e}^{x} + 5 x^{2} y^{2} = 8$,

(ii) $3 x \text{tan} y = 2 x^{2}$.

The equation of a curve is $y^{2} + 4 x^{2} - \text{e}^{y} = 0$.

Find the positive value of $x$ when $y = 0$.

Hence find the gradient of the curve at this point.

The equation of a curve is $2 x y^{2} - x^{2} = 16$. The line $L$ has equation $x = 4$.

Show that the curve has the same gradient at each of the two points where it meets $L$.

State what this result indicates about the tangents to the curve at these points.

The equation of a curve is $3 x \text{e}^{y} + 2 x + 5 = 4 y$.

Verify that the point $( - 1 , 0 )$ lies on the curve and find the equation of the tangent to the curve at $( - 1 , 0 )$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.

The equation of a curve is $\text{ln} y - 2 x y^{3} = 8$.

Show that $\frac{\text{d}y}{\text{d}x} = \frac{2y^{4}}{1-6xy^{3}}$.

Find the equation of the normal to the curve at the point where $y = 1$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.

The equation of a curve is $x y^{2} - 4 x^{2} = 64$.

Show that the stationary points on the curve occur when $x = 4$, and find the exact $y$-coordinates of these stationary points.

The equation of a curve is $\text{ln} ( x y ) + x y^{2} = 1$.

Verify that the point $A ( 1 , 1 )$ lies on the curve.

The tangent to the curve at $A$ meets the $x$-axis at $B$ and the $y$-axis at $C$. Find the area of triangle $O B C$, where $O$ is the origin.

The equation of a curve is $x^{2} y^{2} - 5 x = 22 y$.

Find the gradient of the curve at the point where $x = - 2$ and $y$ is an integer.

The equation of a curve is $3 x^{2} + 2 x y^{3} + 16 = 0$.

Show that the normal to the curve at the point where $x = - 4$ is parallel to the normal to the curve at the point where $x = 4$.

Find the distance between the points where these two normals meet the $y$-axis.

Show that

\dfrac{\text{d}}{\text{d}x}\left(a^{x^{k}}\right) = k a^{x^{k}} x^{k-1}\,\text{ln} \; a

where $a$ and $k$ are constants.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$ for the following

(i) $\text{e}^{xy} + \text{ln} ( x y ) = \text{cosec} x + 4$

(ii) $4 \text{cos} ( x^{2} y ) - 3 \text{e}^{x^{2}y} = 4 \text{e}^{y}$

The equation of a curve is $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ and hence show that the gradient of the curve at any point where it meets a line of the form $y = k x$, where $k \neq 0$, is independent of $x$ and $y$.

The equation of a curve is $\text{ln} y + x^{2} y^{2} = 9$.

Show that the tangents to the curve at the two points where $y = 1$ meet at the point \left(0, \; \dfrac{37}{19}\right).

The equation of a curve is $\text{e}^{\text{sin}(xy)} = 1$, where $y > 0$.

The points $A$ and $B$ have coordinates \left(\dfrac{1}{2}\pi, \; 2\right) and \left(-\dfrac{1}{2}\pi, \; 2\right) respectively.

The tangents to the curve at $A$ and $B$ meet at the point $P$. The tangent at $A$ meets the $x$-axis at $Q$, and the tangent at $B$ meets the $x$-axis at $R$.

Find the area of triangle $P Q R$.