Day 11: Derivatives Review + Quiz

Derivative Definition & Basic Rules (Day 6)Forgot it? Go back and study again →

Limit definition of the derivative

f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}

Tangent / normal line

Tangent:

y = f(a) + f'(a)(x-a)

Normal:

y = f(a) - \frac{1}{f'(a)}(x-a)

Differentiability ⟹ continuity

Differentiable at

a

\implies

continuous at

a

(converse is false:

|x|

at

x=0

)

Fails at corners, cusps, and vertical tangents.

Power, constant, sum/difference

\frac{d}{dx}(x^n) = nx^{n-1}, \quad \frac{d}{dx}(c)=0, \quad \frac{d}{dx}(f \pm g) = f' \pm g'

Product rule

\frac{d}{dx}(fg) = f'g+fg'

Quotient rule

\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g-fg'}{g^2}

Chain Rule, Trig & Inverse Derivatives (Day 7)Forgot it? Go back and study again →

Chain rule

(f \circ g)'(x) = f'(g(x)) \cdot g'(x)

Differentiate the outside, keep the inside, multiply by the inside's derivative.

Six trig derivatives

\frac{d}{dx}\sin x = \cos x, \quad \frac{d}{dx}\cos x = -\sin x, \quad \frac{d}{dx}\tan x = \sec^2 x

\frac{d}{dx}\cot x = -\csc^2 x, \quad \frac{d}{dx}\sec x = \sec x \tan x, \quad \frac{d}{dx}\csc x = -\csc x \cot x

The three 'co-' functions carry the negative sign.

Inverse function theorem

(f^{-1})'(b) = \frac{1}{f'(f^{-1}(b))}

Find the input $a$ with $f(a)=b$ first, then take $\frac{1}{f'(a)}$ .

Inverse trig derivatives

\frac{d}{dx}\sin^{-1}x = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\tan^{-1}x = \frac{1}{1+x^2}, \quad \frac{d}{dx}\sec^{-1}x = \frac{1}{|x|\sqrt{x^2-1}}

$\cos^{-1}, \cot^{-1}, \csc^{-1}$ are just the negatives of their partners above.

Implicit & Exponential/Logarithmic Derivatives (Day 8)Forgot it? Go back and study again →

Implicit differentiation

Differentiate every term w.r.t.

x

; multiply by

\frac{dy}{dx}

whenever you differentiate a

y

e.g. $\frac{d}{dx}(xy) = y+x\frac{dy}{dx}$ , $\frac{d}{dx}(y^3)=3y^2\frac{dy}{dx}$ .

Exponential derivatives

\frac{d}{dx}e^x = e^x, \quad \frac{d}{dx}a^x = a^x \ln a

Logarithmic derivatives

\frac{d}{dx}\ln x = \frac{1}{x}, \quad \frac{d}{dx}\log_a x = \frac{1}{x \ln a}

Logarithmic differentiation

For

y=[f(x)]^{g(x)}

: take

\ln

of both sides,

\ln y = g(x)\ln f(x)

, differentiate implicitly, then multiply by

y

Needed exactly when the variable is in BOTH the base and the exponent.

Rates of Change & Particle Motion (Day 9)Forgot it? Go back and study again →

Marginal cost

C'(n) \approx C(n+1)-C(n)

The approximate cost of one additional unit at production level $n$ .

Position, velocity, acceleration

v(t)=s'(t), \quad a(t)=v'(t)=s''(t), \quad \text{speed}=|v(t)|

Speeding up / slowing down

Same sign (

v,a

)

\implies

speeding up. Opposite signs

\implies

slowing down.

Related Rates & Exponential Growth/Decay (Day 10)Forgot it? Go back and study again →

6-step strategy

1. Draw & label. 2. List known/unknown rates. 3. Write a governing equation. 4. Differentiate w.r.t.

t

. 5. Substitute known values. 6. Solve.

Never substitute a changing value before differentiating.

Common setups

Sphere:

V=\frac{4}{3}\pi r^3

. Cone (similar triangles first):

V=\frac{1}{3}\pi r^2 h

. Right triangle: Pythagorean theorem. Angle of elevation:

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

.

Law of exponential change

\frac{dy}{dt}=ky \implies y=y_0e^{kt}

$k>0$ : growth. $k<0$ : decay.

Half-life

k = -\frac{\ln 2}{T}

, where

T

is the half-life

Newton's Law of Cooling

H(t) = H_s + (H_0-H_s)e^{-kt}

Linearization

L(x) = f(a) + f'(a)(x-a)

, valid for

x

near

a

Differentials

dy = f'(x)\,dx \approx \Delta y

Used to estimate error propagation: measurement error in $x$ becomes $dx$ .

Phase 2 Review

Derivative Definition & Basic Rules (Day 6)Forgot it? Go back and study again →

Chain Rule, Trig & Inverse Derivatives (Day 7)Forgot it? Go back and study again →

Implicit & Exponential/Logarithmic Derivatives (Day 8)Forgot it? Go back and study again →

Rates of Change & Particle Motion (Day 9)Forgot it? Go back and study again →

Phase 2 Quiz: Derivatives