Definition: The derivative of a function f(x)f(x) at a point x=ax = a is the limit: f′(a)=limh→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
The derivative gives the rate of change of the function at a point.
Sum Rule: ddx[f(x)+g(x)]=f′(x)+g′(x)\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)
Difference Rule: ddx[f(x)−g(x)]=f′(x)−g′(x)\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)
Product Rule: ddx[f(x)⋅g(x)]=f′(x)⋅g(x)+f(x)⋅g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)
Quotient Rule: ddx[f(x)g(x)]=f′(x)⋅g(x)−f(x)⋅g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}
Chain Rule is used when differentiating composite functions: ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
If y=f(g(x))y = f(g(x)), then: dydx=f′(g(x))⋅g′(x)\frac{dy}{dx} = f'(g(x)) \cdot g'(x)
Used for functions involving products or quotients of powers.
Take the natural logarithm of both sides of the equation, differentiate, and then solve. y=f(x) ⟹ ln(y)=ln(f(x))y = f(x) \implies \ln(y) = \ln(f(x))
Statement: If a function f(x)f(x) is continuous on [a,b][a, b], differentiable on (a,b)(a, b), and f(a)=f(b)f(a) = f(b), then there exists at least one point c∈(a,b)c \in (a, b) such that f′(c)=0f'(c) = 0.
Statement: If f(x)f(x) is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), then there exists at least one point c∈(a,b)c \in (a, b) such that: f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}
Maclaurin’s Series: Expansion of a function around x=0x = 0: f(x)=f(0)+f′(0)x+f′′(0)2!x2+f(3)(0)3!x3+⋯f(x) = f(0) + f'(0) x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 + \cdots
Taylor’s Series: Expansion of a function around x=ax = a: f(x)=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+⋯f(x) = f(a) + f'(a) (x - a) + \frac{f''(a)}{2!} (x - a)^2 + \cdots
These forms arise when evaluating limits (e.g., 00,∞∞\frac{0}{0}, \frac{\infty}{\infty}).
They often require further techniques such as L'Hôpital’s Rule.
Statement: If the limit results in an indeterminate form 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}, then: limx→af(x)g(x)=limx→af′(x)g′(x)(if the limit exists)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{(if the limit exists)}
Local Maximum: f′(x)=0f'(x) = 0 and f′′(x)<0f''(x) < 0.
Local Minimum: f′(x)=0f'(x) = 0 and f′′(x)>0f''(x) > 0.
Global Maximum/Minimum: Occurs when the function reaches the highest/lowest value over its entire domain.
Involves finding key properties of curves, such as intercepts, asymptotes, concavity, and critical points to sketch the graph of a function.
Leibniz Rule for differentiation of a product: dndxn[f(x)⋅g(x)]=∑k=0n(nk)f(k)(x)⋅g(n−k)(x)\frac{d^n}{dx^n}[f(x) \cdot g(x)] = \sum_{k=0}^{n} \binom{n}{k} f^{(k)}(x) \cdot g^{(n-k)}(x)
This rule provides the nn-th derivative of a product of two functions.
Derivative Computation: Compute the derivative of f(x)=x2sin(x)f(x) = x^2 \sin(x).
Application of Chain Rule: Find the derivative of y=cos(3x2+2)y = \cos(3x^2 + 2).
Rolle's Theorem: Prove that the function f(x)=x2−4x+4f(x) = x^2 - 4x + 4 has at least one point where its derivative is zero in the interval [1,3][1, 3].
Taylor Series: Find the Taylor series expansion of exe^x around x=0x = 0.
Maxima & Minima: Find the local maxima and minima of f(x)=x3−3x+2f(x) = x^3 - 3x + 2.
This is a brief summary of differentiation concepts. Let me know if you need a deeper explanation or help with any of the topics!
Here are some key questions related to Differentiation:
Question: Find the derivative of f(x)=5x2+3x−2f(x) = 5x^2 + 3x - 2.
Question: Find the derivative of f(x)=(2x3+4x)(3x2−5)f(x) = (2x^3 + 4x)(3x^2 - 5) using the product rule.
Question: Differentiate f(x)=sin(3x2+2x)f(x) = \sin(3x^2 + 2x) using the chain rule.
Question: Differentiate f(x)=xxf(x) = x^x using logarithmic differentiation.
Question: Verify Rolle’s Theorem for the function f(x)=x2−4x+4f(x) = x^2 - 4x + 4 in the interval [1,3][1, 3].
Question: Apply the Mean Value Theorem to the function f(x)=x2+2xf(x) = x^2 + 2x on the interval [1,4][1, 4].
Question: Find the Taylor series expansion of f(x)=exf(x) = e^x around x=0x = 0.
Question: Find the local maxima and minima for the function f(x)=x3−6x2+9x+1f(x) = x^3 - 6x^2 + 9x + 1.
Question: Evaluate limx→0sin(x)x\lim_{x \to 0} \frac{\sin(x)}{x} using L'Hôpital's Rule.
Question: Find the second derivative of f(x)=x4−6x3+4x2+xf(x) = x^4 - 6x^3 + 4x^2 + x.
Question: Find the second derivative of f(x)=(x2+1)(x3+2x)f(x) = (x^2 + 1)(x^3 + 2x) using Leibniz’s rule.
Would you like me to solve any of these questions in detail?
The derivative f′(x)f'(x) represents the rate of change of f(x)f(x) with respect to xx.
Formula: f′(x)=limh→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
Power Rule: ddx[xn]=nxn−1\frac{d}{dx}[x^n] = n x^{n-1}
Exponential: ddx[ex]=ex\frac{d}{dx}[e^x] = e^x
Logarithmic: ddx[ln(x)]=1x\frac{d}{dx}[\ln(x)] = \frac{1}{x}
Sum Rule: ddx[f(x)+g(x)]=f′(x)+g′(x)\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)
Product Rule: ddx[f(x)⋅g(x)]=f′(x)⋅g(x)+f(x)⋅g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)
Quotient Rule: ddx[f(x)g(x)]=f′(x)⋅g(x)−f(x)⋅g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}
If y=f(g(x))y = f(g(x)), then: dydx=f′(g(x))⋅g′(x)\frac{dy}{dx} = f'(g(x)) \cdot g'(x)
Useful for complex functions (products/quotients with powers). y=f(x) ⟹ ln(y)=ln(f(x))y = f(x) \implies \ln(y) = \ln(f(x))
Differentiate both sides and solve for dydx\frac{dy}{dx}.
If f(x)f(x) is continuous on [a,b][a, b], differentiable on (a,b)(a, b), and f(a)=f(b)f(a) = f(b), then there exists at least one point c∈(a,b)c \in (a, b) such that f′(c)=0f'(c) = 0.
If f(x)f(x) is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), then there exists at least one point c∈(a,b)c \in (a, b) such that: f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}
Maclaurin Series: Expansion around x=0x = 0: f(x)=f(0)+f′(0)x+f′′(0)2!x2+⋯f(x) = f(0) + f'(0) x + \frac{f''(0)}{2!} x^2 + \cdots
Taylor Series: Expansion around x=ax = a: f(x)=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+⋯f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots
Indeterminate forms: 00,∞∞\frac{0}{0}, \frac{\infty}{\infty}, etc.
L'Hôpital's Rule: limx→af(x)g(x)=limx→af′(x)g′(x)(if the limit exists)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{(if the limit exists)}
Local Maximum: f′(x)=0f'(x) = 0, f′′(x)<0f''(x) < 0
Local Minimum: f′(x)=0f'(x) = 0, f′′(x)>0f''(x) > 0
Global Max/Min: Occurs when the function achieves the highest/lowest value over its entire domain.
For the nn-th derivative of a product of two functions f(x)f(x) and g(x)g(x): dndxn[f(x)⋅g(x)]=∑k=0n(nk)f(k)(x)⋅g(n−k)(x)\frac{d^n}{dx^n}[f(x) \cdot g(x)] = \sum_{k=0}^{n} \binom{n}{k} f^{(k)}(x) \cdot g^{(n-k)}(x)
These are the key concepts in Differentiation for quick reference. Let me know if you need further details or solutions to any specific question!