Differentiation

Exercise 1

Find $f^{\prime}(x)$ for each $f$:

1. $f(x) = \sin(x + x^2)$
2. $f(x) = \sin x + \sin x^2$
3. $f(x) = \sin(\cos x)$
4. $f(x) = \sin(\sin x)$
5. $f(x) = \sin\left(\frac{\cos x}{x}\right)$
6. $f(x) = \frac{\sin(\cos x)}{x}$
7. $f(x) = \sin(x + \sin x)$
8. $f(x) = \sin(\cos(\sin x))$

Exercise 2

Find $f^{\prime}(x)$ for each $f$:

1. $f(x) = \sin((x + 1)^2(x + 2))$
2. $f(x) = \sin^3(x^2 + \sin x)$
3. $f(x) = \sin^2((x + \sin x)^2)$
4. $f(x) = \sin\left(\frac{x^3}{\cos x^3}\right)$
5. $f(x) = \sin(x \sin x) + \sin(\sin x^2)$
6. $f(x) = (\cos x)^3$
7. $f(x) = \sin^2 x \sin x^2 \sin^2 x^2$
8. $f(x) = \sin^3(\sin^2(\sin x))$
9. $f(x) = (x + \sin^5 x)^6$
10. $f(x) = \sin(\sin(\sin(\sin(\sin x))))$
11. $f(x) = \sin((\sin^7 x^7 + 1)^7)$
12. $f(x) = (((x^2 + x)^3 + x)^4 + x)^5$
13. $f(x) = \sin(x^2 + \sin(x^2 + \sin x^2))$
14. $f(x) = \sin(6 \cos(6 \sin(6 \cos 6x)))$
15. $f(x) = \frac{\sin x^2 \sin^2 x}{1 + \sin x}$
16. $f(x) = \frac{1}{x - \frac{2}{x + \sin x}}$
17. $f(x) = \sin \left( \frac{x^3}{\sin \left( \frac{x^3}{\sin x} \right)} \right)$
18. $f(x) = \sin \left( \frac{x}{x - \sin \left( \frac{x}{x - \sin x} \right)} \right)$

Exercise 3

Find derivatives of: tan, cot, sec, csc.

Exercise 4

For each $f$, find $f’(f(x))$:

1. $f(x) = \frac{1}{1 + x}$
2. $f(x) = \sin x$
3. $f(x) = x^2$
4. $f(x) = 17$

Exercise 5

For each $f$, find $f(f’(x))$:

1. $f(x) = \frac{1}{x}$
2. $f(x) = x^2$
3. $f(x) = 17$
4. $f(x) = 17x$

Exercise 6

Find $f’$ in terms of $g’$ if:

1. $f(x) = g(x + g(a))$
2. $f(x) = g(x \cdot g(a))$
3. $f(x) = g(x + g(x))$
4. $f(x) = g(x)(x - a)$
5. $f(x) = g(a)(x - a)$
6. $f(x + 3) = g(x^2)$

Exercise 7

1. A circle with Area: $A=\pi r^2$ has the following: radius 6, $\frac{dr}{dt} = 4$. Find $\frac{dA}{dt}$.
2. A sphere with volume $V=\frac{4}{3} \pi r^3$, $\frac{dr}{dt} = 4$, find $\frac{dV}{dt}$ when $r = 6$.

Exercise 8

Two concentric circles share the same center: area between them is $9\pi$. $\frac{dA_{large}}{dt} = 10\pi$. Find rate of change of smaller circle’s circumference when its area is $16\pi$.

Exercise 9

Particle $A$ moves along x-axis; $B$ moves along $f(x)=-\sqrt{3}x$. At some time:

• A at (5,0), speed 3 units/sec
• B 3 units from origin, speed 4 units/sec

Find rate of change of distance between A and B.

Exercise 10

Given $f(x)=x^{2} \sin \frac{1}{x}$ for $x\neq 0$, $f(0)=0$, and functions $h,k$ with:

$$ \begin{array}{ll} h^{\prime}(x)=\sin^{2}(\sin(x+1)), & h(0)=3 \\ k^{\prime}(x)=f(x+1), & k(0)=0 \end{array} $$

Find:

1. $(f\circ h)^{\prime}(0)$
2. $(k\circ f)^{\prime}(0)$
3. $\alpha^{\prime}(x^{2})$ where $\alpha(x)=h(x^{2})$

Exercise 11

Find $f^{\prime}(0)$ if:

$$ f(x)=\begin{cases} g(x)\sin\frac{1}{x}, & x\neq 0 \\ 0, & x=0 \end{cases} $$

and $g(0)=g^{\prime}(0)=0$.

Exercise 12

Using derivative of $f(x)=1/x$, find $(1/g)^{\prime}(x)$ via Chain Rule.

Exercise 13

1. Find $f^{\prime}(x)$ for $f(x)=\sqrt{1-x^{2}}$ ($-1 < x < 1$).
2. Prove tangent at $(a,\sqrt{1-a^{2}})$ intersects graph only at that point.

Exercise 14

Prove that tangent lines to an ellipse or hyperbola intersect them only once.

Exercise 15

1. If $ f + g $ is differentiable at $ a $, must $ f $ and $ g $ be?
2. If $ f \cdot g $ and $ f $ are differentiable at $ a $, what conditions on $ f $ ensure $ g $ is differentiable at $ a $?

Exercise 16

1. If $ f $ differentiable at $ a $ and $ f(a) \neq 0 $, show $ |f| $ is differentiable at $ a $.
2. Counterexample when $ f(a) = 0 $.
3. If $ f, g $ differentiable at $ a $ and $ f(a) \neq g(a) $, show $ \max(f,g) $ and $ \min(f,g) $ are differentiable at $ a $.
4. Counterexample when $ f(a) = g(a) $.

Exercise 17

Find $ f, g $ where $ g $ is surjective, $ f \circ g $ and $ g $ are differentiable, but $ f $ isn’t.

Exercise 18

1. If $ g = f^2 $, express $ g’ $ in terms of $ f’’ $
2. If $ g = (f’)^2 $, express $ g’ $ in terms of $ f’’’ $
3. If $ f > 0 $ satisfies $ (f’)^2 = f + \frac{1}{f^3} $, find $ f’’’ $ in terms of $ f $

Exercise 19

For $ f $ thrice differentiable with $ f’(x) \neq 0 $, define the Schwarzian derivative:

$$ \Psi f(x) = \frac{f'''(x)}{f'(x)} - \frac{3}{2} \left( \frac{f''(x)}{f'(x)} \right)^2 $$

1. Show $ \Psi(f \circ g) = [\Psi f \circ g] \cdot g’^2 + \Psi g $
2. For $ f(x) = \frac{ax+b}{cx+d} $ ($ ad-bc \neq 0 $), show $ \Psi f = 0 $, hence $ \Psi(f \circ g) = \Psi g $

Exercise 20

Prove Leibniz’s formula:

$$ (f \cdot g)^{(n)}(a) = \sum_{k=0}^n \binom{n}{k} f^{(k)}(a) g^{(n-k)}(a) $$

Exercise 21

If $ f^{(n)}(g(a)) $ and $ g^{(n)}(a) $ exist, prove $ (f \circ g)^{(n)}(a) $ exists (express as sum of products of derivatives).

Exercise 22

1. For $ f(x) = \sum_{k=0}^n a_kx^k $, find $ g $ with $ g’ = f $.
2. For $ f(x) = \sum_{k=2}^m b_k x^{-k} $, find $ g $ with $ g’ = f $.
3. Does $ f(x) = \sum a_kx^k + \sum b_k x^{-k} $ satisfy $ f’(x) = 1/x $?

Exercise 23

Construct degree $ n $ polynomial $ f $ where:

1. $ f’ $ has exactly $ n-1 $ roots
2. $ f’ $ has no roots ($ n $ odd)
3. $ f’ $ has one root ($ n $ even)
4. $ f’ $ has $ k $ roots ($ n-k $ odd)

Exercise 24

1. Prove $ a $ is double root of $ f $ iff $ f(a) = f’(a) = 0 $.
2. When does $ f(x) = ax^2 + bx + c $ ($ a \neq 0 $) have double root? Geometric interpretation?

Exercise 25

For $ f $ differentiable at $ a $, let $ d(x) = f(x) - f’(a)(x-a) - f(a) $. Find $ d’(a) $.

Exercise 26

Given numbers $ a_i, b_i $ and distinct $ x_i $:

1. Find degree $ 2n-1 $ polynomial $ f $ with $ f(x_i) = a_i $, $ f’(x_i) = b_i $, and $ f(x_j) = f’(x_j) = 0 $ for $ j \neq i $.
2. Find degree $ 2n-1 $ polynomial $ f $ with $ f(x_i) = a_i $ and $ f’(x_i) = b_i $ for all $ i $.

Exercise 27

For consecutive roots $ a, b $ of $ f $ (not double roots):

1. Show $ g(a)g(b) > 0 $ where $ f(x) = (x-a)(x-b)g(x) $.
2. Prove $ f’ $ has root in $ (a,b) $.
3. Extend to multiple roots.

Exercise 28

If $ f(x) = xg(x) $ with $ g $ continuous at 0, show $ f $ differentiable at 0 and find $ f’(0) $.

Exercise 29

If $ f $ differentiable at 0 with $ f(0) = 0 $, show $ f(x) = xg(x) $ for $ g $ continuous at 0.

Exercise 30

For $ f(x) = x^{-n} $, prove:

$$ f^{(k)}(x) = (-1)^k \frac{(n+k-1)!}{(n-1)!} x^{-n-k} = (-1)^k k! \binom{n+k-1}{k} x^{-n-k} $$

Exercise 31

Prove $ x \neq f(x)g(x) $ for differentiable $ f,g $ with $ f(0) = g(0) = 0 $.

Exercise 32

Find $ f^{(k)}(x) $ for:

1. $ f(x) = 1/(x-a)^n $
2. $ f(x) = 1/(x^2-1) $

Exercise 33

Let $ f(x) = x^{2n} \sin(1/x) $ ($ f(0) = 0 $). Show $ f^{(k)}(0) $ exist for $ k \leq n $, but $ f^{(n)} $ discontinuous at 0.

Exercise 34

Let $ f(x) = x^{2n+1} \sin(1/x) $ ($ f(0) = 0 $). Show $ f^{(k)}(0) $ exist for $ k \leq n $, $ f^{(n)} $ continuous at 0 but not differentiable there.

Exercise 35

Using Leibniz notation $ \frac{dz}{dx} = \frac{dz}{dy} \frac{dy}{dx} $, find $ \frac{dz}{dx} $ for:

1. $ z = \sin y $, $ y = x + x^2 $
2. $ z = \sin y $, $ y = \cos x $
3. $ z = \sin u $, $ u = \sin x $
4. $ z = \sin v $, $ v = \cos u $, $ u = \sin x $