← Differential Calculus
Mathematics Β· Topic Cheatsheet

Differential Calculus

28 key results accumulated across 3 chapters.

Derivative β€” first principles
Ch 1
fβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
The limit of the secant gradient as the gap h→0h \to 0.
Geometric meaning
Ch 1
fβ€²(x)f'(x) = slope of the tangent to y=f(x)y=f(x) at that point = instantaneous rate of change.
Notation
Ch 1
fβ€²(x)β€…β€Š=β€…β€Šdydxβ€…β€Š=β€…β€ŠyΛ™f'(x) \;=\; \frac{dy}{dx} \;=\; \dot{y}
All three mean the same derivative.
Worked results (from first principles)
Ch 1
x2β†’2x,x3β†’3x2,1xβ†’βˆ’1x2x^2 \to 2x, \quad x^3 \to 3x^2, \quad \tfrac{1}{x} \to -\tfrac{1}{x^2}
Derivative (first principles)
Ch 1
fβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)hf'(x) = \lim_{h\to0}\tfrac{f(x+h)-f(x)}{h}
Meaning
Ch 1
Gradient of the tangent / instantaneous rate of change.
Power rule
Ch 1
ddxxn=nxnβˆ’1\tfrac{d}{dx}x^n = n x^{n-1}
Common trap
Ch 1
Differentiable β‡’ continuous, but not vice-versa (corners/cusps).
Power Rule
Ch 2
ddx[xn]=n x nβˆ’1\frac{d}{dx}\left[x^n\right] = n\,x^{\,n-1}
Any real nn.
Constant multiple
Ch 2
ddx[c f]=c fβ€²\frac{d}{dx}[c\,f] = c\,f'
Sum / difference
Ch 2
ddx[fΒ±g]=fβ€²Β±gβ€²\frac{d}{dx}[f \pm g] = f' \pm g'
Chain Rule
Ch 2
dydx=dyduβ‹…dudx\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}
Derivative of outer (at inner) Γ— derivative of inner.
Product Rule
Ch 2
(fg)β€²=fβ€²g+fgβ€²(fg)' = f'g + fg'
Quotient Rule
Ch 2
(fg)β€²=fβ€²gβˆ’fgβ€²g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}
Common derivatives
Ch 2
ddx[ex]=ex,β€…β€Šβ€…β€Šddx[ln⁑x]=1x,β€…β€Šβ€…β€Šddx[sin⁑x]=cos⁑x,β€…β€Šβ€…β€Šddx[cos⁑x]=βˆ’sin⁑x\frac{d}{dx}[e^x]=e^x,\;\; \frac{d}{dx}[\ln x]=\tfrac{1}{x},\;\; \frac{d}{dx}[\sin x]=\cos x,\;\; \frac{d}{dx}[\cos x]=-\sin x
Product / quotient
Ch 2
(uv)β€²=uβ€²v+uvβ€²,β€…β€Šβ€…β€Š(uv)β€²=uβ€²vβˆ’uvβ€²v2(uv)' = u'v + uv',\;\; \left(\tfrac{u}{v}\right)' = \tfrac{u'v - uv'}{v^2}
Chain rule
Ch 2
dydx=dyduβ‹…dudx\tfrac{dy}{dx} = \tfrac{dy}{du}\cdot\tfrac{du}{dx}
Key derivatives
Ch 2
(sin⁑x)β€²=cos⁑x,β€…β€Š(ex)β€²=ex,β€…β€Š(ln⁑x)β€²=1x(\sin x)'=\cos x,\;(e^x)'=e^x,\;(\ln x)'=\tfrac1x
Common trap
Ch 2
Trig derivatives assume x in RADIANS; don’t forget the chain rule’s inner derivative.
Stationary points
Ch 3
fβ€²(x)=0f'(x) = 0
Maxima, minima, inflexions all have zero gradient.
Second derivative test
Ch 3
fβ€²β€²(x)>0f''(x) > 0 β‡’ minimum (concave up); fβ€²β€²(x)<0f''(x) < 0 β‡’ maximum (concave down).
Increasing / decreasing
Ch 3
fβ€²(x)>0f'(x) > 0 rising; fβ€²(x)<0f'(x) < 0 falling.
Point of inflexion
Ch 3
fβ€²β€²(x)=0f''(x) = 0
Concavity changes sign.
Optimisation method
Ch 3
1) write quantity, 2) reduce to one variable via a constraint, 3) set fβ€²=0f'=0, 4) classify with fβ€²β€²f''.
Stationary points
Ch 3
fβ€²(x)=0f'(x)=0
2nd derivative: f''>0 min, f''<0 max, =0 test further (inflexion?).
Increasing / concave
Ch 3
f'>0 increasing; f''>0 concave up. Point of inflexion where concavity changes.
Optimisation
Ch 3
Write the quantity in ONE variable, differentiate, set =0, justify max/min, check endpoints.
Common trap
Ch 3
f''(x)=0 does NOT guarantee inflexion β€” concavity must actually change sign.