L3 Differentiation
Question Bank
Find the gradient of the normal to the curve $y = 2\sqrt{3x + 4}$ at the point where $x = 4$.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the gradient of the normal to the curve $y = x^2\sin x$ at the point where $x = \dfrac{\pi}{2}$.
Give your answer in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the equation of the tangent to the curve $y = \dfrac{2x - 1}{x + 1}$ at the point where $x = 2$.
Give your answer in the form $y = mx + c$.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the gradient of the normal to the curve $y = \sec(2x)$ at the point where $x = \dfrac{\pi}{6}$.
Give your answer in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the equation of the normal to the curve $y = x\cos(2x)$ at the point where $x = \dfrac{\pi}{4}$.
Give your answer in the form $y = mx + c$, using exact values.
You must use calculus and show any derivatives that you need to find when solving this problem.
The tangent to the curve $y = xe^x$ at the point where $x = 2$ meets the $x$-axis at P.
Find the exact $x$-coordinate of P.
You must use calculus and show any derivatives that you need to find when solving this problem.
The normal to the curve $y = \sqrt{10 - 3x}$ at the point where $x = 2$ meets the $y$-axis at Q.
Find the coordinates of Q.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the equation of the tangent to the curve $y = \sqrt{2x + 3}$ that is parallel to the line $x - 3y = 4$.
Give your answer in the form $ax + by + c = 0$.
You must use calculus and show any derivatives that you need to find when solving this problem.
The point P on the curve $y = \ln(3x - 2)$ has $y$-coordinate $\ln 4$.
Find the equation of the normal to the curve at P. Give your answer in the form $y = mx + c$, in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the equations of the tangents to the curve $y = x^3 - 3x^2 + 2$ that have a gradient of $9$.
You must use calculus and show any derivatives that you need to find when solving this problem.
The normal to the curve $y = \dfrac{8}{x}$ at the point P$(2, 4)$ meets the curve again at Q.
Find the coordinates of Q.
You must use calculus and show any derivatives that you need to find when solving this problem.
The tangent to the curve $y = 4 - \ln x$ at the point where $x = e$ meets the $x$-axis at A and the $y$-axis at B.
Find the exact area of triangle OAB, where O is the origin.
You must use calculus and show any derivatives that you need to find when solving this problem.
The tangent to the curve $y = \dfrac{x - 1}{x + 2}$ at the point P where $x = 1$ is parallel to the tangent at another point Q on the curve.
Find the coordinates of Q and the equation of the tangent at Q, in the form $y = mx + c$.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the equations of the tangents to the curve $y = x + 2\sin x$, $0 \le x \le 2\pi$, that are parallel to the line $y = 2x$.
Give your answers in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
The tangent to the curve $y = \ln x$ at the point where $x = a$ is parallel to the tangent to the curve $y = \dfrac{x^2}{8}$ at the point where $x = a$.
Find the value of $a$, and the equations of both tangents in the form $y = mx + c$.
You must use calculus and show any derivatives that you need to find when solving this problem.
The tangent to the curve $y = e^x - 3x$ at the point where $x = 0$ is perpendicular to the tangent at the point P.
Find the exact coordinates of P.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the equation of every tangent to the curve $y = \sqrt{x + 3}$ that passes through the point $(0, 2)$.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the equation of the straight line that is a tangent to both the curve $y = x^2$ and the curve $y = -\dfrac{1}{x}$.
You must use calculus and show any derivatives that you need to find when solving this problem.