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# Week 8 Problem Set A.
## Reading.
Thomas Ch. 3.6, 3.7 on Chain rule and Implicit differentiation.
**NOTE.** To help you study and check your work, many of these problems come from the relevant sections in Thomas's 12th edition textbook. Find the corresponding problem in the book, and if it is an odd numbered one, the answer is in the back of the book. Of course the numbering below need not match the actual numbering in the book, so you have to look for it.
You got this!
## Problems.
### Derivative calculations.
You need to practice these basic calculations so that they are not going to be a problem to you. This is your new "algebra skill" to master!
For each of the following, compute $\displaystyle \frac{dy}{dx}$:
1. $y=(2x + 1)^{5}$
2. $\displaystyle y=(1- \frac{x}{7})^{-1}$
3. $y=\sec(\tan(x))$
4. $y= \sin^{3}(x)$ .... (Note we have the notation $\sin^{3}(x)= (\sin(x))^{3}$)
For each of the following functions, compute its derivative:
1. $p(t)=\sqrt{3-t}$
2. $\displaystyle s(t)=\frac{4}{3\pi}\sin(3t)+ \frac{4}{5\pi}\cos(5t)$
3. $r(\theta) = (\csc\theta + \cot\theta)^{-1}$
4. $y(x)=x^{2}\sin^{4}(x)+x\cos^{-2}(x)$
5. $\displaystyle y=\frac{1}{21} (3x-2)^{7} + \left(4- \frac{1}{2x^{2}}\right)^{-1}$
6. $y=(4x+3)^{4}(x+1)^{-3}$
7. $h(x)=x\tan(2\sqrt{x})+7$
8. $f(x)=\sqrt{7+x\sec x}$
9. $\displaystyle f(\theta)=\left( \frac{\sin\theta}{1+\cos\theta} \right)^{2}$
10. $r(\theta)=\sin(\theta^{2})\cos(2\theta)$
11. $\displaystyle q(t)=\sin\left( \frac{t}{\sqrt{t+1}} \right)$
In each of the following, find $dy / dt$
1. $y=\sin^{2}(\pi t-2)$
2. $y=(1+\cos(2t))^{-4}$
3. $y=(t \tan t )^{10}$
4. $\displaystyle y = \left( \frac{t^{2}}{t^{3} - 4t} \right)^{3}$
5. $y=\sin(\cos(2t -5))$
6. $\displaystyle y=\left( 1+\tan^{4}\left( \frac{t}{12} \right) \right)^{3}$
7. $y=\sqrt{1+\cos(t^{2})}$
8. $y=\tan^{2}(\sin^{3}(t))$
9. $y=3t(2t^{2}-5)^{4}$
### Second derivatives.
Find $y''$ in each of the following.
1. $\displaystyle y=\left(1+\frac{1}{x}\right)^{3}$
2. $y= \frac{1}{9}\cot(3x-1)$
3. $y = x(2x+1)^{4}$
### Finding derivative values; theory and examples.
1. If $f(u)= u^{5}+1$ and $u=g(x)=\sqrt{x}$, what is $\displaystyle\left.\frac{d}{dx}f(g(x))\right|_{x=1}$?
2. Assume that $f'(3) = -1$, $g'(2) = 5$, $g(2)=3$, and $y=f(g(x))$. What is $y'$ at $x=2$?
3. Find $ds /dt$ when $\theta = 3\pi / 2$ if $s=\cos\theta$ and $d\theta / dt = 5$.
4. Find the tangent line equation to the curve $\displaystyle y=\left( \frac{x-1}{x+1} \right)^{2}$ at $x=0$.
5. Find the tangent line equation to the curve $\displaystyle y=2\tan\left( \frac{\pi x}{4} \right)$ at $x=1$? What is the smallest value the slope of this curve can ever have on the interval $-2 < x < 2$? Explain your answer (Hint: Try to recall the graph of secant...)
### Differentiating implicitly.
Use implicit differentiation to find an expression for $dy/ dx$ in the following:
1. $x^{2}y + xy^{2}= 6$
2. $2xy + y^{2}=x+y$
3. $x^{2}(x-y)^{2}= x^{2}-y^{2}$
4. $\displaystyle y^{2} = \frac{x-1}{x+1}$
5. $x=\tan y$
6. $x+\tan(xy)=0$
7. $y \sin\left( \frac{1}{y} \right) = 1-xy$
Find $dr / d\theta$ in the following:
1. $\theta^{1/2} + r^{1/2}=1$
2. $\displaystyle \sin(r\theta)= \frac{1}{2}$
### Second derivatives.
Use implicit differentiation to find $dy /dx$, and the continue to find the second derivative $d^{2}y /dx^{2}$:
1. $x^{2}+y^{2}=1$
2. $y^{2} = x^{2}+2x$
3. $2\sqrt{y}=x-y$
4. If $x^{3}+y^{3} = 16$, find the value of $d^{2}y / dx^{2}$ at the point $(2,2)$
### Slopes, tangents, and normals.
In the following, first verify the given point $P$ is in the curve $E$, and then **(a)** find the equation of the **tangent line** to $P$, and **(b)** find the equation of the **normal line** at $P$.
1. $E:x^{2}+ xy-y^{2}=1$, $P=(2,3)$
2. $E: x^{2}y^{2}=9$, $P=(3,-4)$
3. $E: 6x^{2}+3xy+2y^{2}+17y - 6 =0$
4. $E: 2xy + \pi \sin y = 2\pi$, $P=(1, \pi / 2)$
5. **The eight curve.** Find the slopes of the curve $y^{4}=y^{2}-x^{2}$ at the two points indicated on the following diagram:![[1 teaching/smc-fall-2023-math-7/week-8/---files/Pasted image 20231019182018.png]]
6. **The devil's curve.** Find the slopes of the devil's curve $y^{4}-4y^{2}=x^{4}-9x^{2}$ at the four indicated points shown below:![[1 teaching/smc-fall-2023-math-7/week-8/---files/Pasted image 20231019182147.png]]