Implicit Differentiation (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

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

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

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

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

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

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

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

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

The equation of a curve is $\text{ln}y=1-xy$. 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 $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

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

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

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

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

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

Use implicit differentiation to show that

$\frac{\text{d}}{\text{d}x}\left(a^x\right)=a^x \text{ln}a$

where $a$ is a constant.

Show that

$\frac{\text{d}}{\text{d}x}\left(a^{kx}\right)=k{a}^{kx} \text{ln}a$

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

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

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

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 $2x{y}^2-x^2=16$. 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 $3x{\text{e}}^y+2x+5=4y$.

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 $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

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

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

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

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

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

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 $OBC$, where $O$ is the origin.

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

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

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

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

$\frac{\text{d}}{\text{d}x}\left(a^{x^k}\right)=k{a}^{x^k}{x}^{k-1} \text{ln}a$

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

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

The 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 $y=kx$, where $k\ne 0$, 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 $\left(0,\frac{37}{19}\right)$.

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

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