Implicit Differentiation (Edexcel IAL Maths): Exam Questions

Find an expression for $\frac{dy}{dx}$ when

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

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

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

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

Find the gradient of the curve with equation $3y^{2}-2x^{3}=10$ at the point (1 , 2).

Show that if $x-\mathrm{sin}y=0$ then $\frac{dy}{dx}=\mathrm{sec}y.$

The curve C has equation

$y^{2}-4x+2=0.$

Show that C intersects the x-axis at the point $( 1 \frac{1}{2} , 0 )$.

(i) Find an expression for $\frac{dy}{dx}$.

(ii) Explain why the curve C does not have any stationary points.

Show that the point $( - \frac{4}{\pi } , \frac{\pi }{2} )$ lies on the curve with equation $2\mathrm{cos}2y=xy$ .

Find an expression for $\frac{dy}{dx}$.

Find the gradient of the curve with equation $12x^{2}-4y^{2}+24=0$at the point (1 , 3).

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

Given $3x^{2}-2y=xy,$find an expression for $\frac{dy}{dx}$.

Hence show that the stationary points of $3x^{2}-2y=xy$ lie on the line $y=6x$.

The curve C has equation $x^{3}+9xy^{2}=54.$

Find the gradient of the tangent to C at the point (3 , 1).

Hence find the equation of the normal to C at (3 , 1), giving your answer in the form $ax+by+c=0$, where *a, b *and c are integers to be found.

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

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

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

Find the gradient of the curve with equation

$3x^{2}y+4x-y=39$

at the point with coordinates (2 , 3).

The curve C has equation

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

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

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

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

Show that if$y=\mathrm{arcsin}x$, then

$\frac{dy}{dx}=\frac{1}{\sqrt{1-x^{2}}}$

Show that the point (0 , π) lies on the curve with equation

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

Find an expression for $\frac{dy}{dx}$.

Find the gradient at the point (0, π).

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

The curve C is defined by the equation $\mathrm{ln}y=1-xy$ .

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

Show that

$\frac{dy}{dx}=-\frac{y^{2}}{1+xy}$

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

Find the equation of the normal to C at point 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 $\frac{dy}{dx}$ in terms of x and y given that

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

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

Hence, or otherwise, find the coordinates of the stationary points.

The curve *C *is given by the equation $e^{xy}=y-x.$

Find the coordinates of the points where *C * intersects the coordinate axes.

Find an expression for $\frac{dy}{dx}$

Show that the tangents to C, at the points where it meets the coordinate axes, have equations

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

The tangents meet at point Q.

Find the distance OQ, where O is the origin.

Use implicit differentiation to show that

d / dx [ a ^ x ] = a ^ x ln a

where a is a constant.

Find an expression for $\frac{dy}{dx}$ in terms of *x *and *y *for the following

(i) $2ye^{x}+5x^{2}y^{2}=8,$

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

Given that

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

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

Hence, or otherwise, find the value of the gradient of

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

at the point where $y=0$ and x is positive.

Show that if $y=\mathrm{arccos}2x$ , then

$\frac{dy}{dx}=-\frac{2}{\sqrt{1-4x^{2}}}.$

The curve C has equation $2xy^{2}-x^{2}=16$, Line L has equation $x=4$.

Show that the two points where C intersects L have equal gradients.

What else can you deduce about the two points where C and L intercept?

Verify that the point (-1 , 0) lies on the curve with equation

$3xe^{y}+2x+5=4y$

and find the equation of the tangent to the curve at the point (-1 , 0).

Give your answer in the form $ax+by+c=0$, where a, b and c are integers to be found.

Show that the derivative function of the curve given by

$\mathrm{ln}y-2xy^{3}=8$

is given by

$\frac{dy}{dx}=\frac{2y^{4}}{1-6xy^{3}}.$

Find the equation of the normal to the curve given in part (a) at the point where $y=1$, giving your answer in the form $ax+by+c=0,$ where $a,b$ and c are integers to be found.

Show that the stationary points on the curve with equation

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

occur when $x=4$, and find the exact y-coordinates of the stationary points.

Verify that the point A(1 , 1) lies on the curve with equation

$\ln ( x y ) + x y^{2} = 1.$

The tangent at point A intercepts the x-axis at point B and the y-axis at point C.

Find the area of the triangle OBC.

Show that

$\frac{ \mathrm{d} }{ \mathrm{d} x} [ a^{k x} ] = k a^{k x} \ln a$

where a and k are constants.

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

(i) $e^{x y} + \ln ( x y ) = \cos e c ( x ) + 4$

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

Find the gradient at the point where$x=-2$ and$y$ is an integer on the curve with equation $x^{2}y^{2}-5x=22y.$

Show that if $2y=\mathrm{arctan}x^{2}$, then

$\frac{dy}{dx}=\frac{x}{1+x^{4}}$

An ellipse has equation

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

Find an expression for $\frac{dy}{dx}$ and hence show that the gradient of the ellipse at any point where it meets a line of the form$y=kx(k\neq0)$ is independent of x and y.

The curve C is described by the equation

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

Show that the tangents of the two points on C where$y=1$meet at the point $( 0 , \frac{37}{19} )$.

The curve C is described by the equation

$3x^{2}+2xy^{3}+16=0.$

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

Find the stationary points and determine their nature for the curve with equation $y^{2}=3x^{2}-2xy+3.$

The curve C is defined by

$e^{\sin x y} = 1 \{ y > 0 \}$

Points *A *and B have coordinates $( \frac{\pi }{2} , 2 )$ and $( - \frac{\pi }{2} , 2 )$ respectively.

The tangents to C at points A and B intersect at the point P.

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

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

Find the area of triangle PQR.

Show that

$\frac{ \mathrm{d} }{ \mathrm{d} x} [ a^{x^{k} } ] = k a^{x^{k} } x^{k - 1} \ln a$

where a and k are constants.