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Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $(ax+b)^m(cx+d)^n$ using the product rule. The answer will be of the form $(ax+b)^{m-1}(cx+d)^{n-1}g(x)$ for a polynomial $g(x)$ . Find $g(x)$ .
Quotient rule
Draft
Question in Bill's workspace by Bill Foster

The derivative of $\displaystyle \frac{ax+b}{\sqrt{cx+d}}$ is $\displaystyle \frac{g(x)}{2(cx+d)^{3/2}}$ . Find $g(x)$ .
Quotient rule
Draft
Question in Bill's workspace by Bill Foster

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$ . Find $g(x)$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate the function $(a + b x)^m e ^ {n x}$ using the product rule.
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $x ^m \sqrt{a x+b}$ .
The answer is in the form $\displaystyle \frac{x^{m-1}g(x)}{2\sqrt{ax+b}}$
for a polynomial $g(x)$ . Find $g(x)$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $f(x) = x^m(a x+b)^n$ .
Product Rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate the function $f(x)=(a + b x)^m e ^ {n x}$ using the product rule. Find $g(x)$ such that $f\;'(x)= (a + b x)^{m-1} e ^ {n x}g(x)$ .
Complete the square. (Video)
Draft
Question in Bill's workspace by Bill Foster

Find $p$ and $q$ such that $ax^2+bx+c = a(x+p)^2+q$ .

Hence, or otherwise, find roots of $ax^2+bx+c=0$ .

Includes a video which shows how to solve a quadratic by completing the square.
Combining algebraic fractions 5.1
Draft
Question in Bill's workspace by Bill Foster

First part: Express $\displaystyle \frac{a}{px + b} +\frac{c}{qx + d},\;a=-c$ . Numerator is an integer.

Second part: $\displaystyle \frac{a}{px + b} +\frac{c}{qx + d}+ \frac{r}{sx+t}$ as single fraction
Combining algebraic fractions 0
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle a \pm \frac{c}{x + d}$ as an algebraic single fraction.
Combining algebraic fractions 3.0
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle b+ \frac{dx+p}{x + q}$ as an algebraic single fraction.
Combining algebraic fractions 3.1
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle ax+b+ \frac{dx+p}{x + q}$ as an algebraic single fraction.
Combining algebraic fractions 3.2
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle \frac{ax+b}{x + c} \pm \frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator.
Combining algebraic fractions 4
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle \frac{ax+b}{cx + d} \pm \frac{rx+s}{px + q}$ as an algebraic single fraction over a common denominator.
Combining algebraic fractions 5.2
Draft
Question in Bill's workspace by Bill Foster

First part: express as a single fraction: $\displaystyle \frac{a}{x + b} + \frac{c}{x + d},\; a \neq -c$ .

Second part: Find $\displaystyle \frac{a}{x + b} + \frac{c}{x + d}+\frac{r}{x+t}$ as a single fraction.
Combining algebraic fractions 5.3
Draft
Question in Bill's workspace by Bill Foster

First part: express as a single fraction: $\displaystyle \frac{a}{px + b} + \frac{c}{qx + d}$ .

Second part: Find $\displaystyle \frac{a}{px + b} + \frac{c}{qx + d}+\frac{r}{sx+t}$ as a single fraction.
Combining algebraic fractions 6.0
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle \frac{a}{x + b} + \frac{cx+d}{x^2 +px+ q}$ as an algebraic single fraction over a common denominator.
Combining algebraic fractions 6.1
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle \frac{a}{x + b} +\frac{c}{(x + b)^2}$ as an algebraic single fraction.
Combining algebraic fractions 6.2
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle \frac{a}{x + b} +\frac{cx+d}{(x + b)^2}$ as an algebraic single fraction.
Combining algebraic fractions 6.3
Draft
Question in Bill's workspace by Bill Foster

Express $\displaystyle \frac{a}{(x+r)(px + b)} + \frac{c}{(x+r)(qx + d)}$ as an algebraic single fraction over a common denominator. The question asks for a solution which has denominator $(x+r)(px+b)(qx+d)$ .
Chain rule
Draft
Question in Bill's workspace by Bill Foster

The derivative of $\displaystyle x ^ {m}(ax^2+b)^{n}$ is of the form $\displaystyle x^{m-1}(ax^2+b)^{n-1}g(x)$ . Find $g(x)$ .
Combining algebraic fractions 1 (Video)
Draft
Question in Bill's workspace by Bill Foster and 1 other

Express $\displaystyle \frac{a}{x + b} \pm \frac{c}{x + d}$ as an algebraic single fraction over a common denominator.

Contains a video in Show steps.
Complete the square
Draft
Question in Bill's workspace by Bill Foster

Find $c$ and $d$ such that $x^2+ax+b = (x+c)^2+d$ .
Combining algebraic fractions 0
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Express $\displaystyle a \pm \frac{c}{x + d}$ as an algebraic single fraction.
Integration by parts
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $\displaystyle \int (ax+b)e^{cx}\;dx =g(x)e^{cx}+C$ , find $g(x)$ . Find $h(x)$ , $\displaystyle \int (ax+b)^2e^{cx}\;dx =h(x)e^{cx}+C$ .
Integration by parts
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx$ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$
Integration by parts
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given that $\displaystyle \int x({ax+b)^{m}} dx=\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$ , find $g(x)$ .
Solve a linear equation
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solve for $x$ : $\displaystyle ax+b = cx+d$
Solve an equation in algebraic fractions
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solve for $x$ : $\displaystyle \frac{px+s}{ax+b} = \frac{qx+t}{cx+d}$ with $pc=qa$ .
Solve an equation with reciprocals
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solve for $x$ : $\displaystyle \frac{s}{ax+b} = \frac{t}{cx+d}$

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