AS-Level Maths | Differentiation
To differentiate $y = ax^n$: multiply the coefficient by the power and subtract one from the power, giving $\displaystyle\frac{dy}{dx} = nax^{n-1}$.
Question 1
Differentiate the following expressions with respect to $x$.
a) $y = x^6 - 7x^2$
b) $y = 1 - 6x^{\frac{5}{2}}$
c) $y = 2x + 8x^{-2}$
a) $\displaystyle\frac{dy}{dx} = 6x^5 - 14x$ b) $\displaystyle\frac{dy}{dx} = -15x^{\frac{3}{2}}$ c) $\displaystyle\frac{dy}{dx} = 2 - 16x^{-3}$
Question 2
Differentiate the following expressions with respect to $x$.
a) $y = x^2 - 4x^6$
b) $y = 5x^3 - 6x^{\frac{3}{2}}$
c) $y = 9x^{-3} + 7x^{-2}$
d) $y = 5 - 5x^{-1}$
e) $y = 7x + \sqrt{x}$
a) $\displaystyle\frac{dy}{dx} = 2x - 24x^5$ b) $\displaystyle\frac{dy}{dx} = 15x^2 - 9x^{\frac{1}{2}}$ c) $\displaystyle\frac{dy}{dx} = -27x^{-4} - 14x^{-3}$ d) $\displaystyle\frac{dy}{dx} = 5x^{-2}$ e) $\displaystyle\frac{dy}{dx} = 7 + \tfrac{1}{2}x^{-\frac{1}{2}}$
To find the x-coordinate of a stationary point: solve $\displaystyle\frac{dy}{dx} = 0$ for $x$.
To find the y-coordinate of a stationary point: substitute $x$ into the equation of the curve.
To determine the nature of a stationary point: substitute $x$ into $\displaystyle\frac{d^2y}{dx^2}$.
If $\displaystyle\frac{d^2y}{dx^2} < 0$ it is a maximum. If $\displaystyle\frac{d^2y}{dx^2} > 0$ it is a minimum.
If $\displaystyle\frac{d^2y}{dx^2} = 0$ check the gradient on either side.
Question 3
For each of the following cubic equations find the coordinates of the stationary points and determine their nature.
a) $y = x^3 - 3x^2 - 9x + 3$
b) $y = x^3 + 12x^2 + 45x + 50$
c) $y = 2x^3 - 6x^2 + 12$
d) $y = 25 - 24x + 9x^2 - x^3$
a) min$(3,\,-24)$, max$(-1,\,8)$ b) min$(-3,\,-4)$, max$(-5,\,0)$ c) min$(2,\,4)$, max$(0,\,12)$ d) min$(2,\,5)$, max$(4,\,9)$
Question 4
For each of the following equations find the coordinates of the stationary points and determine their nature.
a) $\displaystyle y = x + \frac{4}{x},\quad x \neq 0$
b) $\displaystyle y = x^2 + \frac{16}{x},\quad x \neq 0$
c) $y = x - 4\sqrt{x},\quad x > 0$
d) $\displaystyle y = 4x^2 + \frac{1}{x},\quad x \neq 0$
a) min$(2,\,4)$, max$(-2,\,-4)$ b) min$(2,\,12)$ c) min$(4,\,-4)$ d) min$\!\left(\tfrac{1}{2},\,3\right)$
A function is increasing where $\displaystyle\frac{dy}{dx} > 0$, and decreasing where $\displaystyle\frac{dy}{dx} < 0$.
Question 5
For each of the following equations find the range of values of $x$ for which $y$ is increasing or decreasing.
a) $y = 2x^3 - 3x^2 - 12x + 2$, increasing
b) $y = x^3 - 6x^2 + 12$, decreasing
c) $y = x^3 - 3x + 8$, increasing
d) $y = 1 - 3x^2 - x^3$, decreasing
a) $x < -1$ or $x > 2$ b) $0 < x < 4$ c) $x < -1$ or $x > 1$ d) $x < -2$ or $x > 0$
To find the equation of a tangent at a point: use $\displaystyle\frac{dy}{dx}$ to find the gradient, substitute $x$ into the curve to find the $y$-coordinate, then use $y = mx + c$.
Question 6
For each of the following curves find an equation of the tangent at the point whose $x$-coordinate is given.
a) $y = x^2 - 9x + 13$, where $x = 6$
b) $y = x^4 + x + 1$, where $x = 1$
c) $y = 2x^2 + 6x + 7$, where $x = -1$
d) $y = 2x^3 - 4x + 5$, where $x = 1$
e) $y = 2x^3 - 4x^2 - 3$, where $x = 2$
a) $y = 3x - 23$ b) $y = 5x - 2$ c) $y = 2x + 5$ d) $y = 2x + 1$ e) $y = 8x - 19$
To find the equation of a normal at a point: use $\displaystyle\frac{dy}{dx}$ to find the gradient, take the negative reciprocal, substitute $x$ into the curve to find the $y$-coordinate, then use $y = mx + c$.
Question 7
For each of the following curves find an equation of the normal at the point whose $x$-coordinate is given.
a) $f(x) = x^3 - 4x^2 + 1$, where $x = 2$
b) $f(x) = x^3 - 7x^2 + 11x$, where $x = 3$
c) $f(x) = 3x^4 - 7x^3 + 5$, where $x = 2$
d) $f(x) = \tfrac{1}{4}x^5 - 18x + 11$, where $x = 2$
a) $4y = x - 30$ b) $4y = x - 15$ c) $12y + x + 34 = 0$ d) $2y + x + 32 = 0$
Modelling basics
Question 8
Find the rate of change of each quantity with respect to the stated variable.
a) If $A = \pi x^2 - 20x$, find the rate of change of $A$ with respect to $x$.
b) If $V = x - 2\pi x^3$, find the rate of change of $V$ with respect to $x$.
c) If $P = at^2 - bt$, find the rate of change of $P$ with respect to $t$.
d) If $W = 6kh^{\frac{1}{2}} - h$, find the rate of change of $W$ with respect to $h$.
e) If $N = (at + b)^2$, find the rate of change of $N$ with respect to $t$.
a) $\displaystyle\frac{dA}{dx} = 2\pi x - 20$ b) $\displaystyle\frac{dV}{dx} = 1 - 6\pi x^2$ c) $\displaystyle\frac{dP}{dt} = 2at - b$ d) $\displaystyle\frac{dW}{dh} = 3kh^{-\frac{1}{2}} - 1$ e) $\displaystyle\frac{dN}{dt} = 2a^2t + 2ab$
Solving harder equations
Question 9
For each of the following equations find the coordinates of the stationary points and determine their nature.
a) $\displaystyle y = \frac{1}{x} - \frac{1}{\sqrt{x}},\quad x > 0$
b) $\displaystyle y = \frac{3\sqrt{x} - 2}{x^{\frac{3}{2}}},\quad x > 0$
c) $\displaystyle y = \sqrt[3]{x} + \frac{27}{x},\quad x > 0$
d) $\displaystyle y = \frac{1}{3\sqrt{x}}\!\left(\frac{2}{x} - 3\right),\quad x > 0$
a) min$\!\left(4,\,-\tfrac{1}{4}\right)$ b) max$(1,\,1)$ c) min$(27,\,4)$ d) min$\!\left(2,\,-\displaystyle\frac{\sqrt{2}}{3}\right)$
Differentiation from first principles
The derivative from first principles is $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.
Question 10
Prove each of the following from first principles.
a) The derivative of $4x$ is $4$.
b) The derivative of $x^3$ is $3x^2$.
c) The derivative of $2x^3$ is $6x^2$.
d) The derivative of $5x^2$ is $10x$.
e) The derivative of $kx^3$ is $3kx^2$, where $k$ is a constant.
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