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

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

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

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

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$y=\arcsin x$, then 

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

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

$3\tan y=2xy$

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 $\ln y=1-xy$ .

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

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

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

$\frac{\mathrm{d}}{\mathrm{dx}}\left[a^x\right]=a^x\ln a$

where a is a constant.

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

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

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

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 $y=\arccos 2x$ , then

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

The curve C has equation $2x{y}^2-x^2=16$, 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

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

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 

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

is given by

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

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

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

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

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

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

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

where a and k are constants.

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

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

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

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 $2y=\arctan x^2$, 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$y=kx(k\ne 0)$ is independent of x and y.

The curve C is described by the equation

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

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

The curve C is described by the 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 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

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

Points A and B have coordinates $\left(\frac{\pi }{2},2\right)$ and $\left(-\frac{\pi }{2},2\right)$ 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

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

where a and k are constants.