True or false: \(\ds\diff{}{x}\{f(x)+g(x)\}=f'(x)+g'(x)\) when \(f\) and \(g\) are differentiable functions.
2
True or false: \(\ds\diff{}{x}\{f(x)g(x)\}=f'(x)g'(x)\) when \(f\) and \(g\) are differentiable functions.
3
True or false: \(\ds\diff{}{x}\left\{\dfrac{f(x)}{g(x)}\right\}=\dfrac{f'(x)}{g(x)}-\dfrac{f(x)g'(x)}{g^2(x)}\) when \(f\) and \(g\) are differentiable functions.
4
Let \(f\) be a differentiable function. Use at least three different rules to differentiate \(g(x)=3f(x)\text{,}\) and verify that they all give the same answer.
Exercises — Stage 2
5
Differentiate \(f(x)=3x^2+4x^{1/2}\) for \(x\gt 0\text{.}\)
6
Use the product rule to differentiate \(f(x)=(2x+5)(8\sqrt{x}-9x)\text{.}\)
7(✳)
Find the equation of the tangent line to the graph of \(y=x^3\) at \(x=\dfrac{1}{2}\text{.}\)
8(✳)
A particle moves along the \(x\)--axis so that its position at time \(t\) is given by \(x=t^3-4t^2+1\) .
At \(t=2\text{,}\) what is the particle's speed?
At \(t=2\text{,}\) in what direction is the particle moving?
At \(t=2\text{,}\) is the particle's speed increasing or decreasing?
9(✳)
Calculate and simplify the derivative of \(\dfrac{2x-1}{2x+1}\)
10
What is the slope of the graph \(y=\left(\dfrac{3x+1}{3x-2}\right)^2\) when \(x=1\text{?}\)
11
Find the equation of the tangent line to the curve \(f(x)=\dfrac{1}{\sqrt{x}+1}\) at the point \(\left(1,\frac{1}{2}\right)\text{.}\)
Exercises — Stage 3
12
A town is founded in the year 2000. After \(t\) years, it has had \(b(t)\) births and \(d(t)\) deaths. Nobody enters or leaves the town except by birth or death (whoa). Give an expression for the rate the population of the town is growing.
13(✳)
Find all points on the curve \(y=3x^2\) where the tangent line passes through \((2,9)\text{.}\)
14(✳)
Evaluate \(\displaystyle \lim_{y\rightarrow 0}\left(
\dfrac{\sqrt{100180+y}-\sqrt{100180}}{y}\right)\) by interpreting the limit as a derivative.
15
A rectangle is growing. At time \(t=0\text{,}\) it is a square with side length 1 metre. Its width increases at a constant rate of 2 metres per second, and its length increases at a constant rate of 5 metres per second. How fast is its area increasing at time \(t \gt 0\text{?}\)
16
Let \(f(x)=x^2g(x)\) for some differentiable function \(g(x)\text{.}\) What is \(f'(0)\text{?}\)
17
Verify that differentiating \(f(x)=\dfrac{g(x)}{h(x)}\) using the quotient rule gives the same answer as differentiating \(f(x)=\dfrac{g(x)}{k(x)}\cdot\dfrac{k(x)}{h(x)}\) using the product rule and the quotient rule.