Singapore GCE O-Level · Additional Mathematics

Differentiation and Applications

69 practice questions, each with the answer and an explanation.

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Find the derivative of $y = \frac{l n x}{x ^{2}}$ with respect to $x$ for $x > 0$ .
The equation of a curve is $y = 2 x^{3} - 9 x^{2} + 12 x + 5$ . Find the coordinates of the stationary points and determine their nature.
A spherical balloon is being inflated so that its volume increases at a constant rate of $36 π cm^{3} / s$ . Find the rate of increase of its radius when the radius is $3 cm$ .
Find the equation of the normal to the curve $y = 3 e^{2 x} - 1$ at the point where $x = 0$ .
Find the derivative of $f (x) = x^{2} cos (3 x)$ .
A piece of wire of length $40 cm$ is bent into the shape of a sector of a circle of radius $r cm$ and angle $θ$ radians. Find the value of $r$ that maximizes the enclosed area of th…
Find the gradient of the curve $y = 3 x^{3} - 4 x^{2} + 5 x$ at the point where $x = 2$ .
Find the gradient of the curve $y = 2 x^{3} + 5 x^{2} - x$ at the point where $x = 1$ .
Find the gradient of the curve $y = 4 x^{3} - 2 x^{2} + 7 x$ at the point where $x = 3$ .
Find the gradient of the curve $y = x^{3} + 6 x^{2} - 3 x$ at the point where $x = 2$ .
Find the gradient of the curve $y = 5 x^{3} - x^{2} + 2 x$ at the point where $x = 1$ .
Find the gradient of the curve $y = 2 x^{3} - 8 x^{2} + 4 x$ at the point where $x = 3$ .
Find the gradient of the curve $y = 3 x^{3} + 3 x^{2} + 3 x$ at the point where $x = 2$ .
Find the gradient of the curve $y = 6 x^{3} - 5 x^{2} + x$ at the point where $x = 1$ .
Differentiate $y = (2 x + 1)^{3}$ and find the gradient when $x = 1$ .
Differentiate $y = (3 x - 1)^{4}$ and find the gradient when $x = 0$ .
Differentiate $y = (2 x + 3)^{4}$ and find the gradient when $x = 1$ .
Differentiate $y = (5 x - 2)^{3}$ and find the gradient when $x = 1$ .
Differentiate $y = (3 x + 1)^{5}$ and find the gradient when $x = 0$ .
Differentiate $y = (2 x - 3)^{5}$ and find the gradient when $x = 1$ .
Find the gradient of the curve $y = x^{2} (3 x + 1)$ at $x = 1$ .
Find the gradient of the curve $y = x^{3} (2 x + 4)$ at $x = 1$ .
Find the gradient of the curve $y = x^{2} (5 x - 2)$ at $x = 2$ .
Find the gradient of the curve $y = x^{3} (1 x + 3)$ at $x = 1$ .
Find the gradient of the curve $y = x^{4} (2 x - 1)$ at $x = 1$ .
Find the gradient of the curve $y = x^{2} (4 x + 1)$ at $x = 1$ .
Find the $x$ -coordinates of the stationary points of the curve $y = 2 x^{3} - 12 x^{2} + 18 x$ .
Find the $x$ -coordinates of the stationary points of the curve $y = 2 x^{3} - 18 x^{2} + 48 x$ .
Find the $x$ -coordinates of the stationary points of the curve $y = 2 x^{3} - 18 x^{2} + 30 x$ .
Find the $x$ -coordinates of the stationary points of the curve $y = 2 x^{3} - 24 x^{2} + 90 x$ .
Find the $x$ -coordinates of the stationary points of the curve $y = 2 x^{3} - 6 x^{2} - 18 x$ .
Find the $x$ -coordinates of the stationary points of the curve $y = 2 x^{3} - 24 x^{2} + 72 x$ .
The curve $y = 2 x^{3} - 12 x^{2} + 18 x + 5$ has two stationary points. Find the $y$ -coordinate of the maximum point.
The curve $y = 2 x^{3} - 21 x^{2} + 60 x + 5$ has two stationary points. Find the $y$ -coordinate of the maximum point.
The curve $y = 2 x^{3} - 15 x^{2} + 24 x + 5$ has two stationary points. Find the $y$ -coordinate of the maximum point.
The curve $y = 2 x^{3} + 3 x^{2} - 12 x + 5$ has two stationary points. Find the $y$ -coordinate of the maximum point.
The curve $y = 2 x^{3} - 21 x^{2} + 36 x + 5$ has two stationary points. Find the $y$ -coordinate of the maximum point.
Find the equation of the tangent to the curve $y = x^{2} + 2 x$ at the point where $x = 1$ .
Find the equation of the tangent to the curve $y = x^{2} + 1 x$ at the point where $x = 2$ .
Find the equation of the tangent to the curve $y = x^{2} + 1 x$ at the point where $x = 3$ .
Find the equation of the tangent to the curve $y = x^{2} + 3 x$ at the point where $x = 2$ .
Find the equation of the tangent to the curve $y = x^{2} + 4 x$ at the point where $x = 1$ .
Find the equation of the tangent to the curve $y = x^{2} + 2 x$ at the point where $x = - 1$ .
Find the gradient of the normal to the curve $y = x^{2} + 3 x$ at the point where $x = 1$ .
Find the gradient of the normal to the curve $y = x^{2} + 2 x$ at the point where $x = 2$ .
Find the gradient of the normal to the curve $y = x^{2} + 5 x$ at the point where $x = 1$ .
Find the gradient of the normal to the curve $y = x^{2} + 1 x$ at the point where $x = 3$ .
Differentiate $y = e^{2 x}$ and find the gradient at $x = 0$ .
Differentiate $y = e^{3 x}$ and find the gradient at $x = 1$ .
Differentiate $y = e^{2 x}$ and find the gradient at $x = 1$ .
Differentiate $y = e^{4 x}$ and find the gradient at $x = 0$ .
Differentiate $y = e^{5 x}$ and find the gradient at $x = 1$ .
Find the gradient of $y = ln (2 x + 1)$ at $x = 0$ .
Find the gradient of $y = ln (3 x + 1)$ at $x = 0$ .
Find the gradient of $y = ln (1 x + 2)$ at $x = 1$ .
Find the gradient of $y = ln (2 x + 3)$ at $x = 1$ .
Find the gradient of $y = ln (4 x + 1)$ at $x = 0$ .
Find the gradient of $y = 2 sin 1 x$ at $x = 0$ .
Find the gradient of $y = 3 sin 2 x$ at $x = 0$ .
Find the gradient of $y = 4 sin 3 x$ at $x = 0$ .
Find the gradient of $y = 5 sin 1 x$ at $x = 0$ .
The radius of a circle is increasing at a rate of $2$ cm/s. Find the rate of increase of the area when the radius is $3$ cm. Leave your answer in terms of $π$ .
The radius of a circle is increasing at a rate of $2$ cm/s. Find the rate of increase of the area when the radius is $5$ cm. Leave your answer in terms of $π$ .
The radius of a circle is increasing at a rate of $2$ cm/s. Find the rate of increase of the area when the radius is $6$ cm. Leave your answer in terms of $π$ .
The radius of a circle is increasing at a rate of $2$ cm/s. Find the rate of increase of the area when the radius is $10$ cm. Leave your answer in terms of $π$ .
The side of a cube is increasing at a rate of $0.5$ cm/s. Find the rate of increase of its volume when the side is $4$ cm.
The side of a cube is increasing at a rate of $0.5$ cm/s. Find the rate of increase of its volume when the side is $5$ cm.
The side of a cube is increasing at a rate of $0.5$ cm/s. Find the rate of increase of its volume when the side is $7$ cm.
The side of a cube is increasing at a rate of $0.5$ cm/s. Find the rate of increase of its volume when the side is $8$ cm.

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