Introduction to Differentiation
Definition: Differentiation refers to the process of finding the derivative of a
function. The derivative represents the rate of change of the function's output
with respect to its input. If f(x)f(x)f(x) is a function, its derivative f′(x)f'(x)f′(x) or
dfdx\frac{df}{dx}dxdf measures how f(x)f(x)f(x) changes as xxx changes.
Conceptual Understanding:
Geometric Interpretation: The derivative at a point x=ax = ax=a is the
slope of the tangent line to the graph of the function at that point.
Physical Interpretation: In physics, the derivative can represent
instantaneous velocity if f(x)f(x)f(x) represents position with respect to
time.
2. Basic Rules of Differentiation
1. Power Rule: If f(x)=xnf(x) = x^nf(x)=xn, where nnn is a constant, then the
derivative f′(x)=nxn−1f'(x) = n x^{n-1}f′(x)=nxn−1.
2. Constant Rule: If f(x)=cf(x) = cf(x)=c, where ccc is a constant, then f′
(x)=0f'(x) = 0f′(x)=0.
3. Constant Multiple Rule: If f(x)=c⋅g(x)f(x) = c \cdot g(x)f(x)=c⋅g(x), where
ccc is a constant, then f′(x)=c⋅g′(x)f'(x) = c \cdot g'(x)f′(x)=c⋅g′(x).
4. Sum Rule: If f(x)=g(x)+h(x)f(x) = g(x) + h(x)f(x)=g(x)+h(x), then f′(x)=g′(x)
+h′(x)f'(x) = g'(x) + h'(x)f′(x)=g′(x)+h′(x).
5. Difference Rule: If f(x)=g(x)−h(x)f(x) = g(x) - h(x)f(x)=g(x)−h(x), then f′
(x)=g′(x)−h′(x)f'(x) = g'(x) - h'(x)f′(x)=g′(x)−h′(x).
6. Product Rule: If f(x)=g(x)⋅h(x)f(x) = g(x) \cdot h(x)f(x)=g(x)⋅h(x), then f′
(x)=g′(x)⋅h(x)+g(x)⋅h′(x)f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)f′(x)=g′(x)⋅h(x)
+g(x)⋅h′(x).
7. Quotient Rule: If f(x)=g(x)h(x)f(x) = \frac{g(x)}{h(x)}f(x)=h(x)g(x), then f′
(x)=g′(x)⋅h(x)−g(x)⋅h′(x)[h(x)]2f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}
{[h(x)]^2}f′(x)=[h(x)]2g′(x)⋅h(x)−g(x)⋅h′(x).
8. Chain Rule: If f(x)=g(h(x))f(x) = g(h(x))f(x)=g(h(x)), then f′(x)=g′(h(x))⋅h′
(x)f'(x) = g'(h(x)) \cdot h'(x)f′(x)=g′(h(x))⋅h′(x).
3. Derivatives of Common Functions
1. Exponential Functions:
ddxex=ex\frac{d}{dx} e^x = e^xdxdex=ex
ddxax=axln(a)\frac{d}{dx} a^x = a^x \ln(a)dxdax=axln(a), where aaa is
a positive constant.
2. Logarithmic Functions:
ddxln(x)=1x\frac{d}{dx} \ln(x) = \frac{1}{x}dxdln(x)=x1
ddxloga(x)=1xln(a)\frac{d}{dx} \log_a(x) = \frac{1}{x \ln(a)}dxdloga
(x)=xln(a)1, where aaa is a positive constant.
3. Trigonometric Functions:
ddxsin(x)=cos(x)\frac{d}{dx} \sin(x) = \cos(x)dxdsin(x)=cos(x)
ddxcos(x)=−sin(x)\frac{d}{dx} \cos(x) = -\sin(x)dxdcos(x)=−sin(x)
ddxtan(x)=sec2(x)\frac{d}{dx} \tan(x) = \sec^2(x)dxdtan(x)=sec2(x)
ddxcot(x)=−csc2(x)\frac{d}{dx} \cot(x) = -\csc^2(x)dxdcot(x)=−csc2(x)
ddxsec(x)=sec(x)tan(x)\frac{d}{dx} \sec(x) = \sec(x) \tan(x)dxd
sec(x)=sec(x)tan(x)
ddxcsc(x)=−csc(x)cot(x)\frac{d}{dx} \csc(x) = -\csc(x) \cot(x)dxd
csc(x)=−csc(x)cot(x)
4. Inverse Trigonometric Functions:
ddxarcsin(x)=11−x2\frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1 -
x^2}}dxdarcsin(x)=1−x21
ddxarccos(x)=−11−x2\frac{d}{dx} \arccos(x) = -\frac{1}{\sqrt{1 -
x^2}}dxdarccos(x)=−1−x21
ddxarctan(x)=11+x2\frac{d}{dx} \arctan(x) = \frac{1}{1 + x^2}dxd
arctan(x)=1+x21
ddx\arccot(x)=−11+x2\frac{d}{dx} \arccot(x) = -\frac{1}{1 + x^2}dxd\
arccot(x)=−1+x21
ddx\arcsec(x)=1∣x∣x2−1\frac{d}{dx} \arcsec(x) = \frac{1}{|x| \sqrt{x^2 -
1}}dxd\arcsec(x)=∣x∣x2−11
ddx\arccsc(x)=−1∣x∣x2−1\frac{d}{dx} \arccsc(x) = -\frac{1}{|x| \
sqrt{x^2 - 1}}dxd\arccsc(x)=−∣x∣x2−11
4. Higher-Order Derivatives
Definition: The second derivative f′′(x)f''(x)f′′(x) is the derivative of the first
derivative f′(x)f'(x)f′(x). It measures the curvature or concavity of the function.
Notation:
f′′(x)f''(x)f′′(x) or d2fdx2\frac{d^2 f}{dx^2}dx2d2f for the second
derivative.
Higher-order derivatives are denoted as f(n)(x)f^{(n)}(x)f(n)(x) or dnfdxn\
frac{d^n f}{dx^n}dxndnf.
5. Applications of Differentiation
1. Finding Local Extrema:
Critical Points: Points where f′(x)=0f'(x) = 0f′(x)=0 or f′(x)f'(x)f′(x) is
undefined.
First Derivative Test: Determines if a critical point is a local maximum,
minimum, or neither based on the sign of f′(x)f'(x)f′(x) around the point.
Second Derivative Test: Uses f′′(x)f''(x)f′′(x) to determine concavity. If f′′
(x)>0f''(x) > 0f′′(x)>0, the function is concave up (local minimum); if f′′
(x)<0f''(x) < 0f′′(x)<0, the function is concave down (local maximum).
2. Optimization Problems:
Used to find maximum or minimum values of functions subject to
constraints.
3. Related Rates:
Involves finding the rate at which one quantity changes with respect to
another.
4. Curve Sketching:
Derivatives help analyze the shape of a graph by determining intervals of
increase or decrease, concavity, and points of inflection.
6. Implicit Differentiation
When a function is given implicitly (e.g., x2+y2=1x^2 + y^2 = 1x2+y2=1),
implicit differentiation involves differentiating both sides of the equation with
respect to xxx and solving for dydx\frac{dy}{dx}dxdy.
Example: Given x2+y2=1x^2 + y^2 = 1x2+y2=1: Differentiate implicitly:
2x+2ydydx=02x + 2y \frac{dy}{dx} = 02x+2ydxdy=0 Solve for dydx\frac{dy}
{dx}dxdy: dydx=−xy\frac{dy}{dx} = -\frac{x}{y}dxdy=−yx
7. Differentiability and Continuity
A function must be continuous at a point to be differentiable there, but continuity
alone does not guarantee differentiability. Differentiability implies continuity.
Key Points:
A function can be continuous at a point but not differentiable (e.g.,
f(x)=∣x∣f(x) = |x|f(x)=∣x∣ at x=0x = 0x=0).
A function differentiable at a point is also continuous at that point.
8. Techniques for Differentiation
1. Differentiating Parametric Equations:
Given x=f(t)x = f(t)x=f(t) and y=g(t)y = g(t)y=g(t), the derivative dydx\
frac{dy}{dx}dxdy is found using dydt\frac{dy}{dt}dtdy and dxdt\
frac{dx}{dt}dtdx: dydx=dydtdxdt\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\
frac{dx}{dt}}dxdy=dtdxdtdy
2. Differentiating Using Logarithmic Differentiation:
Useful for functions of the form y=f(x)g(x)y = f(x)^{g(x)}y=f(x)g(x). Take
the natural logarithm of both sides, differentiate, then solve for dydx\
frac{dy}{dx}dxdy.
3. Differentiating Implicit Functions:
For functions defined implicitly, use implicit differentiation to find
derivatives.
