Here’s a compact but comprehensive set of Calculus I (Differential Calculus) notes —
perfect for a quick study or crash revision session. If you're looking for something even more
visual or step-by-step with examples, just let me know!
📘 Calculus I Summary Notes
1. Limits and Continuity
Limit of a function:
limx→af(x)=L\lim_{x \to a} f(x) = L
Means as xx gets close to aa, f(x)f(x) gets close to LL.
Basic Limit Laws:
o lim(f⋅g)=limf⋅limg\lim (f \cdot g) = \lim f \cdot \lim g
o lim(f+g)=limf+limg\lim (f + g) = \lim f + \lim g
o lim(fg)=limflimg\lim \left( \frac{f}{g} \right) = \frac{\lim f}{\lim g}, limg≠0\
lim g \ne 0
Continuity at x=ax = a:
A function is continuous at x=ax = a if:
f(a)f(a) is defined
limx→af(x)\lim_{x \to a} f(x) exists
limx→af(x)=f(a)\lim_{x \to a} f(x) = f(a)
2. Derivatives
Definition:
f′(x)=limh→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
Geometrically: Slope of the tangent line at a point.
Physically: Instantaneous rate of change.
3. Basic Derivative Rules
ddx(c)=0\frac{d}{dx}(c) = 0
ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}
ddx(cf(x))=cf′(x)\frac{d}{dx}(cf(x)) = c f'(x)
ddx(f+g)=f′+g′\frac{d}{dx}(f + g) = f' + g'
Product Rule:
(fg)′=f′g+fg′(fg)' = f'g + fg'
Quotient Rule:
(fg)′=f′g−fg′g2\left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}
Chain Rule:
ddxf(g(x))=f′(g(x))⋅g′(x)\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)
4. Common Derivatives
ddx(sinx)=cosx\frac{d}{dx}(\sin x) = \cos x
ddx(cosx)=−sinx\frac{d}{dx}(\cos x) = -\sin x
ddx(tanx)=sec2x\frac{d}{dx}(\tan x) = \sec^2 x
ddx(ex)=ex\frac{d}{dx}(e^x) = e^x
ddx(lnx)=1x\frac{d}{dx}(\ln x) = \frac{1}{x}
5. Applications of Derivatives
Critical Points: Where f′(x)=0f'(x) = 0 or is undefined.
First Derivative Test:
If f′f' changes from + to –, it's a local max.
If f′f' changes from – to +, it's a local min.
Second Derivative Test:
If f′′(x)>0f''(x) > 0, local min
If f′′(x)<0f''(x) < 0, local max
Concavity & Inflection Point:
f′′>0f'' > 0: Concave up
f′′<0f'' < 0: Concave down
Inflection point where concavity changes.
Optimization:
Find max/min values using derivative tests.
6. Related Rates
Differentiate both sides of an equation with respect to time tt.
Use chain rule where needed.
Example: If A=πr2A = \pi r^2, then
dAdt=2πrdrdt\frac{dA}{dt} = 2\pi r \frac{dr}{dt}
7. Implicit Differentiation
When yy is a function of xx, but not isolated:
Differentiate both sides and use chain rule:
Example: x2+y2=1⇒2x+2ydydx=0\text{Example: } x^2 + y^2 = 1 \Rightarrow 2x + 2y\
frac{dy}{dx} = 0
Solve for dydx\frac{dy}{dx}.
8. Mean Value Theorem (MVT)
If ff is continuous on [a, b] and differentiable on (a, b),
∃ c∈(a,b) such that f′(c)=f(b)−f(a)b−a\exists\ c \in (a, b) \text{ such that } f'(c) = \frac{f(b) -
f(a)}{b - a}
9. Rolle’s Theorem
Special case of MVT:
If f(a)=f(b)f(a) = f(b), then
\exists\ c \in (a, b) \text
