# Basic Differentiation

# Basic Differentiation

## Definition and Fundamental Theorem

**Differentiation**is a fundamental operation in calculus that produces a function showing the rate of change of one quantity compared to another.- The principal concept behind differentiation is the notion of the
**limit**. - The
**Fundamental Theorem of Calculus**states that differentiation and integration are inverse processes.

## Power Rule

- The
**Power Rule**for differentiation states that if y = x^n, its derivative is given by dy/dx = n*x^(n-1). - This is applicable to any real number n (n ≠ -1).

## Constant, Sum and Difference Rule

- The derivative of a
**constant**is zero. - The derivative of the
**sum**or**difference**between two functions is the sum or difference of their derivatives.

## Product and Quotient Rule

- The
**Product Rule**: the derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. - The
**Quotient Rule**: the derivative of the quotient of two functions is the bottom function times the derivative of the top function minus the top function times derivative of the bottom function, all over the bottom function squared.

## Chain Rule

- The
**Chain Rule**: When differentiating composite functions, the derivative of the outer function is multiplied by the derivative of the inner function.

## Implicit Differentiation

**Implicit differentiation**is used when it is difficult to solve an equation for y explicitly as a function of x.- The technique is to differentiate each side of the equation with respect to x, treating y as a function of x, and consequently applying the chain rule when necessary.

## Higher Derivatives

- Second or
**higher derivatives**can be found by differentiating the derivative of a function. - The second derivative often represents information about the geometry of the function, for instance, whether the function is concave up or down.

## Applications of Derivatives

- Derivatives have various applications in real-world situations, such as physics, engineering, economics, etc.
- In many problems involving rates of change or maximization/minimization problems, differentiation is involved.
- The derivative at a point can be used to find the
**tangent**to the curve at that point and the**normal**, which is perpendicular to the tangent.