Differentiation: Mathematics 9709 (Cambridge International AS & A Level)

A curve has equation $y = 4x^3 - \frac{6}{x^2} + 8\sqrt{x}.$

Which of the following is $\dfrac{dy}{dx}$?

A curve $C$ has equation $y = (2x - 3)^3$.

(a) Use the chain rule to find $\dfrac{dy}{dx}$. [2]

(b) Find the gradient of $C$, and the equation of the tangent to $C$, at the point where $x = 2$. [3]

(c) Find the equation of the normal to $C$ at the point where $x = 2$, giving your answer in the form $ax + by = c$, where $a$, $b$ and $c$ are integers. [3]

A curve has equation $y = x^3 - 6x^2 + 9x + 2$.

(a) Find $\dfrac{dy}{dx}$. [2]

(b) Find the coordinates of the two stationary points on the curve. [4]

(c) By considering $\dfrac{d^2y}{dx^2}$ at each stationary point, determine whether it is a maximum or a minimum point. [2]

(d) State the set of values of $x$ for which the curve is a decreasing function. [2]

A spherical soap bubble is expanding. Its radius is $r$ cm at time $t$ seconds, and its volume is $V$ cm³. The radius increases at a constant rate of $0.3$ cm per second.

(a) Write down an expression for $V$ in terms of $r$, and find $\dfrac{dV}{dr}$. [2]

(b) Using the chain rule, find $\dfrac{dV}{dt}$ in terms of $r$. [2]

(c) Find the rate of increase of the volume, in cm³ per second, at the instant when $r = 5$. Give your answer both as a multiple of $\pi$ and correct to $3$ significant figures. [2]

(d) At a later instant, the volume is increasing at a rate of $4.8\pi$ cm³ per second. Find the radius of the bubble at this instant. [2]

A curve has equation $y = x^3 - 4x$. When $x = 3$, $x$ increases by a small amount $\delta x = 0.02$.

Using differentiation, which of the following is the best approximation for the corresponding small increase in $y$, $\delta y$?

A curve has equation $y = 5x^4 - \frac{2}{x^3} + 6\sqrt[3]{x}.$

Which of the following is $\dfrac{dy}{dx}$?

A curve $C$ has equation $y = \sqrt{2x + 7}$, for $x > -\dfrac{7}{2}$.

(a) Use the chain rule to find $\dfrac{dy}{dx}$. [2]

(b) Find the gradient of $C$, and the equation of the tangent to $C$, at the point where $x = 1$, giving the tangent in the form $ay = bx + c$, where $a$, $b$ and $c$ are integers. [3]

(c) Find the equation of the normal to $C$ at the point where $x = 1$, giving your answer in the form $px + y = q$, where $p$ and $q$ are integers. [3]

A curve has equation $y = 2x^3 - 3x^2 - 12x + 5$.

(a) Find $\dfrac{dy}{dx}$. [2]

(b) Find the coordinates of the two stationary points on the curve. [4]

(c) By considering $\dfrac{d^2y}{dx^2}$ at each stationary point, determine whether it is a maximum or a minimum point. [2]

(d) State the set of values of $x$ for which the curve is an increasing function. [2]

The side length of a cube is increasing at a constant rate of $2$ cm per second.

Using differentiation, which of the following is the rate of increase of the cube's total surface area, in cm² per second, at the instant when the side length is $5$ cm?

A curve has equation $y = x^2 + \dfrac{16}{x}$, for $x \neq 0$.

(a) Find $\dfrac{dy}{dx}$. [2]

(b) Find the coordinates of the stationary point of the curve, and use $\dfrac{d^2y}{dx^2}$ to determine whether it is a maximum or a minimum point. [4]

(c) Find the equation of the tangent to the curve at the point where $x = 4$. [3]