## What are Derivatives?

Derivatives are referred to the situation when a quantity **p** varies with respect to another quantity **q, **fulfilling a condition p =f (q) and dp/dq (or f ‘ (q)) represents the change of rate of **p** with respect to **q **and dp/dq where q = \(q_{0}\) or f ‘ (\(q_{0}\)) represents the rate of change of **p **with respect to **q** at q = \(q_{0}\).

### Chain Rule

Let us consider two variables **p** and **q **which vary with respect to the presence of another variable** r, **which means **p** = f(**r**) and **q** = g(**r**), then according to the chain rule,

### Increasing and Decreasing Function

A Function **p **can be said increasing when:

- We can see an increase on an interval (a,b) if \(x_{1}<x_{2}\) in (a,b) =>\(p(x_{1})<p(x_{2})\) for all \(x_{1}, x_{2}\epsilon (a,b)\)

A function is said to be decreasing when:

- if \(x_{1}<x_{2}\) in (a,b) when: \(x_{1}>x_{2}\) for all \(x_{1}, x_{2}\epsilon (a,b)\)

### Constant

The constant term in (a,b) if p(x)= q for all x \epsilon (a,b)\) and q is a constant

### Equation of Tangent

The equation of a tangent is given at a point \((x_{0}, y_{0})\) to the curve y =f(x) is representated by

y-y_{o}=dy/dx]_{(xo,yo)}(x-x_{o})

### Critical point

The point **q **inside the domain of a function **f** at which f ‘ (**q**) =0 or **f **cannot be differentiable is referred to as the critical point of **f.**

### Important Questions

- Find out the value of m if line y=mx+1 which is a tangent to the curve \(y^{2}=4x\).
- For the curve \(2y+x^{2}=3\) is ?

- A semicircular opening is surmounted on a window which is in the form of a rectangle The total perimeter of the window is 50 m. Find the dimensions of the window to admit maximum light through the whole opening.

- A point on the hypotenuse of a triangle is at distance p and q from the sides of the triangle.

- For what values of a the function f given by f(x) = \(x^{2}\) + ax + 1 is increasing On [1, 2]?

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