Implicit Differentiation (Edexcel A Level Maths: Pure): Exam Questions

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A curve has equation

$x^3+2xy+3y^2=47$

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.

The point $P\left(-2,5\right)$ lies on the curve.

Find the equation of the normal to the curve at $P$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be found.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$ where appropriate, given that

(i) $2xy+y^2=4$

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

The curve $C$ has equation

$3x^{2}y+4x-y=41$

The point $P(2,3)$ lies on $C$.

Find the exact value of the gradient of $C$ at the point $P$.

The curve $C$ has equation

$\frac{1}{5}{x}^2{\text{e }}^y=5$

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

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

Hence find the gradients of $C$ at the two points where $C$ intersects the $x$-axis.

Given that

$y=\text{arcsin}x$

show that

$\frac{\text{d}y}{\text{d}x}=\frac{1}{\sqrt{1-x^2}}$

The curve $C$ has equation

$3\text{tan}y=2xy$

Show that the point $P(0,\pi )$ lies on $C$.

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

Find the exact value of the gradient of $C$ at the point $P$.

Hence find an equation of the tangent to $C$ at the point $P$.

The curve $C$ has equation

$\text{ln}y=1-xy$

The point $P(1,1)$ lies on $C$.

Show that

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

Find the gradient of the tangent to $C$ at the point $P$, and hence find the gradient of the normal to $C$ at $P$.

Find an equation of the normal to $C$ at the point $P$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be found.

The curve $C$ has equation

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

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

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

Hence, or otherwise, find the exact coordinates of the stationary points on $C$.

The curve $C$ has equation

${\text{e}}^{xy}=y-x$

Find the coordinates of the points where $C$ crosses the coordinate axes.

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

Show that the tangents to $C$ at the points where it crosses the coordinate axes have equations

$y=2x+1$ and $2y=x+1$

The two tangents meet at the point $Q$.

Find the exact distance $OQ$, where $O$ is the origin.

Given that

$y=a^x$

where $a$ is a positive constant, use implicit differentiation to show that

$\frac{\text{d}y}{\text{d}x}=a^x\text{ln}a$

Figure 8 shows a sketch of the curve $C$ with equation $y=x^x,x>0$.

Find, by firstly taking logarithms, the $x$ coordinate of the turning point of $C$.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

The curve $C$ has equation

$x^2\tan y=90<y<\frac{\pi }{2}$

Show that

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{-18x}{x^4+81}$

Prove that $C$ has a point of inflection at $x=\sqrt[4]{27}$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$, given that

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

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

A curve has equation

$2x^3+y^2-3xy=7$

Show that

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{3y-6x^2}{2y-3x}$

Find the equation of the normal to the curve at the point $P(2,3)$.

The curve $C$ has equation

$y^2+4x^2-{\text{e}}^y=0$

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

Hence, or otherwise, find the value of the gradient of $C$ at the point where $y=0$ and $x$ is positive.

Given that

$y=\text{arccos}2x$

show that

$\frac{\text{d}y}{\text{d}x}=-\frac{2}{\sqrt{1-4x^2}}$

The curve $C$ has equation

$2x{y}^2-x^2=16$

The line $l$ has equation $x=4$.

Show that the gradient of $C$ is the same at both points where $C$ intersects $l$.

State what else can be deduced about these two points of intersection.

The curve $C$ has equation

$3x{\text{e}}^y+2x+5=4y$

Verify that the point $P(-1,0)$ lies on $C$, and find an equation of the tangent to $C$ at the point $P$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be found.

The curve $C$ has equation

$\text{ln}y-2x{y}^3=8$

Show that

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

Find an equation of the normal to $C$ at the point where $y=1$. Give your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be found.

The curve $C$ has equation

$x{y}^2-4x^2=64$

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

A curve has equation

$\text{ln}\left(xy\right)+x{y}^2=1$

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

The tangent to the curve at the point $A$ intersects the $x$-axis at the point $B$ and the $y$-axis at the point $C$.

Find the exact area of the triangle $OBC$, where $O$ is the origin.

Given that

$y=a^{kx}$

where $a$ and $k$ are constants with $a>0$, use implicit differentiation to show that

$\frac{\text{d}y}{\text{d}x}=k a^{kx}\text{ln}a$

The curve $C$ has equation

$p{x}^3+qxy+3y^2=26$

where $p$ and $q$ are constants.

Show that

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{ap{x}^2+bqy}{qx+cy}$

where $a$, $b$ and $c$ are integers to be found.

Given that

• the point $P\left(-1,-4\right)$ lies on $C$

• the normal to $C$ at $P$ has equation $19x+26y+123=0$

find the value of $p$ and the value of $q$.

Figure 4 shows a sketch of the curve with equation $x^2-2xy+3y^2=50$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y-x}{3y-x}$.

The curve is used to model the shape of a cycle track with both $x$ and $y$ measured in km.

The points $P$ and $Q$ represent points that are furthest west and furthest east of the origin $O$, as shown in Figure 4.

Using part (a), find the exact coordinates of the point $P$.

Explain briefly how to find the coordinates of the point that is furthest north of the origin $O$. (You do not need to carry out this calculation).

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$, given that

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

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

The curve $C$ has equation

$x^2{y}^2-5x=22y$

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

Given that

$2y=\text{arctan}\left(x^2\right)$

show that

$\frac{\text{d}y}{\text{d}x}=\frac{x}{1+x^4}$

The curve $C$ has equation

$\frac{x^2}{4}+\frac{y^2}{9}=1$

Find an expression for $\frac{\text{d}y}{\text{d}x}$ and hence show that the gradient of $C$ at any point where it meets the line $y=kx$, where $k$ is a non-zero constant, is independent of $x$ and $y$.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has equation

$\text{ln}y+x^2{y}^2=9$

Show that the tangents to $C$ at the points where $y=1$ intersect at the point $\left(0,\frac{37}{19}\right)$.

The curve $C$ has equation

$3x^2+2x{y}^3+16=0$

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

Find the exact distance between the $y$-axis intercepts of these two normals.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has equation

$y^2=3x^2-2xy+3$

Find the exact coordinates of the stationary points on $C$ and determine their nature.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has equation

${\text{e}}^{\text{sin}\left(xy\right)}=1\{y>0\}$

The points $A\left(\frac{\pi }{2},2\right)$ and $B\left(-\frac{\pi }{2},2\right)$ lie on $C$.

The tangent to $C$ at $A$ and the tangent to $C$ at $B$ intersect at the point $P$.

The tangent to $C$ at $A$ intersects the $x$-axis at the point $Q$.

The tangent to $C$ at $B$ intersects the $x$-axis at the point $R$.

Find the exact area of triangle $PQR$.

Given that

$y=a^{x^k}$

where $a$ and $k$ are constants with $a>0$, use implicit differentiation to show that

$\frac{\text{d}y}{\text{d}x}=k a^{x^k}{x}^{k-1}\text{ln}a$