The Basics of Differentiation
- Theorem 1: Sum Rule
- Theorem 2: Product Rule
- Theorem 3: Quotient Rule
- Theorem 4: Chain Rule
- Theorem 5: Power Rule
Constant Rule
Mean Value Theorem
Applications of Differentiation
FAQs on Theorems On Differentiation

Theorems On Differentiation

Differentiation is one of the most important concepts in calculus, helping us understand how things change. It’s used in many areas like physics, economics, and engineering. Let’s explore the key theorems in differentiation with examples and easy explanations.

The Basics of Differentiation

Differentiation helps us find how a function changes at any given point. For example, it can tell us how fast a car is moving if we know its position over time.

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The derivative of a function $f(x)$ $f$ $($ $x$ $)$ is written as:

f'(x) \text{ or } \frac{df}{dx}

$f$ $'$ $($ $x$ $)$ $or$ $dxdf$ $$

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The formula to calculate it is:

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

$f$ $'$ $($ $x$ $)$ $=$ $h$ $\to$ $0$ $lim$ $$ $hf$ $($ $x$ $+$ $h$ $)$ $-$ $f$ $($ $x$ $)$ $$

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Theorem 1: Sum Rule

The Sum Rule states that the derivative of a sum of two functions is the sum of their derivatives:

\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)

$dxd$ $$ $[$ $f$ $($ $x$ $)$ $+$ $g$ $($ $x$ $)]$ $=$ $f$ $'$ $($ $x$ $)$ $+$ $g$ $'$ $($ $x$ $)$

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Example: If $f(x) = x^2$ $f$ $($ $x$ $)$ $=$ $x$ $2$ and $g(x) = 3x$ $g$ $($ $x$ $)$ $=$ $3$ $x$ , then:

\frac{d}{dx}[x^2 + 3x] = \frac{d}{dx}[x^2] + \frac{d}{dx}[3x] = 2x + 3

$dxd$ $$ $[$ $x$ $2$ $+$ $3$ $x$ $]$ $=$ $dxd$ $$ $[$ $x$ $2$ $]$ $+$ $dxd$ $$ $[$ $3$ $x$ $]$ $=$ $2$ $x$ $+$ $3$

Theorem 2: Product Rule

The Product Rule is for finding the derivative of the product of two functions:

\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

$dxd$ $$ $[$ $f$ $($ $x$ $)$ $\cdot$ $g$ $($ $x$ $)]$ $=$ $f$ $'$ $($ $x$ $)$ $g$ $($ $x$ $)$ $+$ $f$ $($ $x$ $)$ $g$ $'$ $($ $x$ $)$

Example: If $f(x) = x^2$ $f$ $($ $x$ $)$ $=$ $x$ $2$ and $g(x) = \sin x$ $g$ $($ $x$ $)$ $=$ $sin$ $x$ , then:

\frac{d}{dx}[x^2 \sin x] = (2x)(\sin x) + (x^2)(\cos x) = 2x\sin x + x^2\cos x

$dxd$ $$ $[$ $x$ $2$ $sin$ $x$ $]$ $=$ $($ $2$ $x$ $)$ $($ $sin$ $x$ $)$ $+$ $($ $x$ $2$ $)$ $($ $cos$ $x$ $)$ $=$ $2$ $x$ $sin$ $x$ $+$ $x$ $2$ $cos$ $x$

Theorem 3: Quotient Rule

The Quotient Rule helps when functions are divided:

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

$dxd$ $$ $[$ $g$ $($ $x$ $)$ $f$ $($ $x$ $)$ $$ $]$ $=$ $[$ $g$ $($ $x$ $)]$ $2$ $f$ $'$ $($ $x$ $)$ $g$ $($ $x$ $)$ $-$ $f$ $($ $x$ $)$ $g$ $'$ $($ $x$ $)$ $$

Example: If $f(x) = x^2$ $f$ $($ $x$ $)$ $=$ $x$ $2$ and $g(x) = x+1$ $g$ $($ $x$ $)$ $=$ $x$ $+$ $1$ , then:

\frac{d}{dx}\left[\frac{x^2}{x+1}\right] = \frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2}

$dxd$ $$ $[$ $x$ $+$ $1$ $x$ $2$ $$ $]$ $=$ $($ $x$ $+$ $1$ $)$ $2$ $($ $2$ $x$ $)$ $($ $x$ $+$ $1$ $)$ $-$ $($ $x$ $2$ $)$ $($ $1$ $)$ $$

Theorem 4: Chain Rule

The Chain Rule is used for composite functions:

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

$dxd$ $$ $[$ $f$ $($ $g$ $($ $x$ $))]$ $=$ $f$ $'$ $($ $g$ $($ $x$ $))$ $\cdot$ $g$ $'$ $($ $x$ $)$

Example: If $f(u) = u^2$ $f$ $($ $u$ $)$ $=$ $u$ $2$ and $g(x) = \sin x$ $g$ $($ $x$ $)$ $=$ $sin$ $x$ , then:

\frac{d}{dx}[\sin^2 x] = 2(\sin x)(\cos x)

$dxd$ $$ $[$ $sin$ $2$ $x$ $]$ $=$ $2$ $($ $sin$ $x$ $)$ $($ $cos$ $x$ $)$

Theorem 5: Power Rule

The Power Rule is simple and widely used:

\frac{d}{dx}[x^n] = n \cdot x^{n-1}

$dxd$ $$ $[$ $x$ $n$ $]$ $=$ $n$ $\cdot$ $x$ $n$ $-$ $1$

Example: If $f(x) = x^3$ $f$ $($ $x$ $)$ $=$ $x$ $3$ , then:

\frac{d}{dx}[x^3] = 3x^2

$dxd$ $$ $[$ $x$ $3$ $]$ $=$ $3$ $x$ $2$

Constant Rule

The derivative of a constant is zero:

\frac{d}{dx}[c] = 0

$dxd$ $$ $[$ $c$ $]$ $=$ $0$

Example: If $f(x) = 5$ $f$ $($ $x$ $)$ $=$ $5$ , then:

\frac{d}{dx}[5] = 0

$dxd$ $$ $[$ $5$ $]$ $=$ $0$

Mean Value Theorem

This theorem says if a function is continuous and differentiable, there is at least one point where the derivative equals the average rate of change:

f'(c) = \frac{f(b) - f(a)}{b - a}

$f$ $'$ $($ $c$ $)$ $=$ $b$ $-$ $af$ $($ $b$ $)$ $-$ $f$ $($ $a$ $)$ $$

Applications of Differentiation

Physics: Calculating speed and acceleration.\n
Economics: Understanding profit and cost changes.
Biology: Studying population growth rates.

FAQs on Theorems On Differentiation

In actual life, what role could differentiation play?

Differentiation's value in everyday life cannot be overstated. It is used to solve a variety of calculations in everyday life. It is used to determine the maximum and lowest values of particular quantities known as functions, such as cost, profit, and loss.

What is the Leibnitz theorem for integral state differentiation?

There will be an integral sign for differentiation in the Leibnitz theorem. The product of two functions can be differentiated up to the nth order, and this can be stated using the formula.

What distinguishes derivatives?

Differentiation is the procedure for calculating derivatives. The derivative of the function y = f(x), where x represents the rate at which the value of y changes when variable x changes.

The Basics of Differentiation
- Theorem 1: Sum Rule
- Theorem 2: Product Rule
- Theorem 3: Quotient Rule
- Theorem 4: Chain Rule
- Theorem 5: Power Rule
Constant Rule
Mean Value Theorem
Applications of Differentiation
FAQs on Theorems On Differentiation

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