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# Week 7 Problem Set B.
## Reading.
Thomas Chapter 3.1, 3.2, 3.3, 3.4, 3.5.
Preview chapter 3.6 for next week as well -- it is about one of the most import rule in derivative calculations: The chain rule.
### Binomial coefficients, Pascal triangles, and some basic combinatorics.
1. Without doing too much work, write out the expansion to the binomial expression $(a+b)^{6}$
2. Without doing too much work, write out the expansion to the binomial expression $(a+3b)^{4}$
3. Without doing too much work, write out the expansion to the binomial expression $(2a-b)^{3}$
4. You go to an ice cream shop that features 20 different flavors. How many ways can you create an ice cream delight with 3 flavors of your choosing?
### Derivative rule practice: Power, sum, product, and quotient rule, trigonometric functions.
Practice computing basic derivatives. We will speak of the **second derivative** of $f(x)$ to mean taking the derivative of the derivative of $f(x)$, that is, $$
f''(x)= \frac{d}{dx}\left( \frac{d}{dx} f(x)\right)
$$and in general the third derivative as $f'''(x)$, and even higher derivative as $f^{(n)}(x)$.
1. In each of the following, compute the first and second derivatives.
1. $y=-x^{2}+3$
2. $s=5t^{3}-3t^{5}$
3. $\displaystyle y= \frac{4x^{3}}{3}-x$
4. $\displaystyle w=3z^{-2}-\frac{1}{z}$
5. $y=6x^{2}-10x-5x^{-2}$
6. $\displaystyle r= \frac{1}{3s^{2}} - \frac{5}{2s}$
2. In each of the following, find $y'$ using the product rule.
1. $y = (3-x^{2})(x^{3}-x+1)$
2. $\displaystyle y=(x^{2}+1)\left( x+5 + \frac{1}{x} \right)$
3. Find the derivatives of the functions below:
1. $\displaystyle y= \frac{2x+5}{3x-2}$
2. $\displaystyle g(x) = \frac{x^{2}-4}{x+0.5}$
3. $v= (1-t)(1+t^{2})$
4. $\displaystyle f(s)= \frac{\sqrt{s}-1}{\sqrt{s}+1}$
5. $\displaystyle v= \frac{1+x-4\sqrt{x}}{x}$
6. $\displaystyle y = \frac{1}{(x^{2}-1)(x^{2}+x+1)}$
4. Write down the derivatives of the trigonometric functions:
1. $\frac{d}{dx}\sin(x)$
2. $\frac{d}{dx}\cos(x)$
3. $\frac{d}{dx} \tan(x)$
4. $\frac{d}{dx}\sec(x)$
5. $\frac{d}{dx} \csc(x)$
6. $\frac{d}{dx}\cot(x)$
5. Find $\frac{dy}{dx}$ for the following:
1. $y =x^{2}\cos(x)$
2. $y=\sqrt{x}\sec(x)+3$
3. $y=\sin(x)\tan(x)$
4. $y=(\sec(x)+\tan(x))(\sec(x)-\tan(x))$
5. $\displaystyle y= \frac{\cot(x)}{1+\cot(x)}$
### Repeated derivatives.
Again, we write $f'',f''', ...$ and $f^{(n)}$ in general for repeated differentiation $n$ times. Equivalently we also write $$\frac{d^{n}}{dx^{n}}f(x)=f^{(n)}(x)$$
1. Consider the polynomial $f(x)=5x^{3}-2x^{2}+x+1$. Compute $f',f'',f''',f'''$. What do you observe?
2. If $f(x)$ is a degree $n$ polynomial, on which derivative $m$ would $f^{(m)}(x)$ be identically zero?
3. Consider $y=\sin(x)$. Write down $y,y',y'',y''',y^{(4)},y^{(5)},y^{(6)}, y^{(7)}$. What do you observe?
4. What is $y^{(423)}$ if $y=\sin(x)$?
### Proofs of the derivative rules.
In class we stated and proved these basic derivatives. Review the notes, or read the text. Learn their proofs and write them down here. The proofs all involve using the limit definition of derivative. It'll be good to know how to do these.
1. Prove the constant rule: $$
\frac{d}{dx}(C)=0
$$
2. Prove the power rule, when $n$ is a positive integer: $$
\frac{d}{dx}(x^{n})=nx^{n-1}
$$(note, the power rule is actually true for $n$ any real numbers)
3. Prove the constant multiple rule: If $f$ is differentiable, and $C$ is a constant, then $$
\frac{d}{dx}(Cf(x))=Cf'(x)
$$
4. Prove the sum rule: If $f$ and $g$ are differentiable, then $$
\frac{d}{dx}(f(x)+g(x))=f'(x)+g'(x)
$$
5. Prove the product rule: If $f$ and $g$ are differentiable, then $$
\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)
$$
6. Prove the quotient rule: If $f$ and $g$ are differentiable, and $g\neq 0$, then $$
\frac{d}{dx}\left( \frac{f(x)}{g(x)} \right) = \frac{g(x)f'(x)-f(x)g'(x)}{g(x)^{2}}
$$
7. Prove the following derivatives of sine and cosine: $$
\frac{d}{dx}(\sin(x))=\cos(x) \quad , \frac{d}{dx}(\cos(x))=-\sin(x)
$$Recall the limits $\displaystyle\lim_{h\to 0} \frac{\sin(h)}{h}=1$ and $\displaystyle\lim_{h\to 0 } \frac{1-\cos(h)}{h}=0$, and the angle sum formulas for sine and cosine.
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