Differentiation – Derivative
In mathematics derivative is about rates of change. The way a tree’s height changes according to passing years or the way the car’s speed changes is its acceleration.
A derivative represents the rate at which a function changes with respect to one of its variables. Derivative measures how the function’s output changes in response to a change in its input. The derivative is also called the gradient function.
Gradient is slope of a line. Slope of a line is given by formula.
Average rate of change of a function is given by formula.
If function is linear than average rate of change is equal to slope of line.
Derivative is the rate of change of function on a point. Derivative is the rate of change in an instant. To find rate of change of a function at an instant or a point can be calculated by using limit.
The formula above is the definition of derivative in mathematical way. Another formula is given by
Both formulas describe instantaneous rate of change.
The gradient of the function at the point P is equal to the gradient of the tangent at point P. The gradient of the function changes as x changes. The derivative of a function varies as x changes. By using formulas above or using rules of derivative the gradient of the function at a point P can be calculated.
The derivative of the function f with respect to c can be written as
Derivatives have numerous applications across various fields, including physics, engineering, economics, and computer science. They are fundamental in calculus and play crucial role in understanding the behavior of functions, optimizing processes, and solving real-world problems.
Differentiation is the process of finding an expression of the derivative from the expression of a function.
There are many rules for derivative.
Power rule 
, 
Product rule 
Quotient Rule 
, 
Chain Rule 
, 
Derivative of composition functions 
, 
Derivative of inverse of functions
Derivative of trigonometric functions
Derivative of inverse trigonometric functions
Derivative of exponential and logarithmic functions
Derivative of Implicit functions
As we mentioned before derivative is the gradient of functions. If derivative is positive at a certain point then function is increasing. If derivative is negative at a certain point than function is decreasing.
By using higher order derivative, second order derivative, concavity of a function can be identified. If second derivative is positive then function is concave up. If second derivative is negative the function is concave down.
In IB Math AA syllables Calculus, derivative, differentiations are seen at
IB Math SL AA Topic 5 Differentiation and Rules of Derivative
IB Math HL AA Topic 5 Differentiation and Rules of Derivative
EduIB Questionbank gives you opportunity to solve real past paper like questions about derivative.
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For example one of differentiation – rules of derivative questions in EduIB is shown below. It is very similar to IB Math SL Exam in May 2014.
