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What is differentiation?

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Differentiation is the process of finding the rate at which a quantity changes with respect to another.

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What is differentiation?

Differentiation is the process of finding the rate at which a quantity changes with respect to another.

What is the definition of the derivative?

The derivative of a function $f(x)$ is defined as: $$ f'(x) = rac{d}{dx} f(x) = 	ext{lim}_{h \to 0} \frac{f(x+h) - f(x)}{h} $$.

What is the Power Rule in differentiation?

The Power Rule states that $$ \frac{d}{dx} [x^n] = nx^{n-1} $$.

What is the Sum Rule in differentiation?

The Sum Rule states that $$ \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) $$.

What is the Product Rule in differentiation?

The Product Rule states that $$ \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) $$.

What is the Quotient Rule in differentiation?

The Quotient Rule states that $$ \frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2} $$.

What is the Chain Rule in differentiation?

The Chain Rule states that $$ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) $$.

What is a second derivative?

The second derivative is the derivative of the first derivative, denoted as $$ f''(x) = \frac{d}{dx} [f'(x)] $$.

What is the purpose of finding maxima and minima in calculus?

Maxima and minima are found at critical points where $$ f'(x) = 0 $$, and the second derivative test determines their nature.

What is integration?

Integration is the reverse process of differentiation, used to find the area under curves.

What is the definition of an indefinite integral?

An indefinite integral represents a family of functions and is expressed as: $$ \int f(x) \, dx = F(x) + C $$.

What is the Power Rule in integration?

The Power Rule states that $$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) $$.

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus states that $$ \int_{a}^{b} f(x) \, dx = F(b) - F(a) $$, where $F(x)$ is the antiderivative of $f(x)$.

What is a definite integral?

A definite integral of $f(x)$ from $a$ to $b$ is expressed as: $$ \int_{a}^{b} f(x) \, dx $$.

What is a differential equation?

A differential equation is an equation involving derivatives of a function, expressed as $$ \frac{dy}{dx} = f(x, y) $$.

What are separable equations in differential equations?

Separable equations can be expressed as $$ \frac{dy}{dx} = g(x)h(y) $$, allowing separation of variables for integration.

What is the integrating factor in linear differential equations?

The integrating factor for a linear equation $$ \frac{dy}{dx} + P(x)y = Q(x) $$ is $$ \mu(x) = e^{\int P(x) \, dx} $$.

What is the importance of calculus in JEE Main?

Calculus is a fundamental component of the JEE Main Mathematics syllabus, essential for success in the exam.