Find from first principle, the derivative of $2 x^2-x$ with respect to $x$.
Answer (2)
The derivative of the function 2 x 2 − x with respect to x using first principles is 4 x − 1 . We used the limit definition of the derivative to derive this result step-by-step. The simplification and evaluation process leads us to the final answer without any errors.
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Define the derivative from first principles as f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x ) .
Substitute f ( x ) = 2 x 2 − x into the definition: f ′ ( x ) = lim h → 0 h 2 ( x + h ) 2 − ( x + h ) − ( 2 x 2 − x ) .
Simplify the expression by expanding, combining like terms, and canceling h .
Evaluate the limit to find the derivative: 4 x − 1 .
Explanation
Problem Setup We are asked to find the derivative of the function f ( x ) = 2 x 2 − x with respect to x using the first principle definition of the derivative.
Definition of Derivative The first principle definition of the derivative is given by: f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x )
Substitution Now, we substitute f ( x ) = 2 x 2 − x into the definition: f ′ ( x ) = h → 0 lim h 2 ( x + h ) 2 − ( x + h ) − ( 2 x 2 − x )
Expansion Next, we expand the terms in the numerator: f ′ ( x ) = h → 0 lim h 2 ( x 2 + 2 x h + h 2 ) − x − h − 2 x 2 + x f ′ ( x ) = h → 0 lim h 2 x 2 + 4 x h + 2 h 2 − x − h − 2 x 2 + x
Simplification Now, we simplify the numerator: f ′ ( x ) = h → 0 lim h 4 x h + 2 h 2 − h
Factoring We factor out h from the numerator: f ′ ( x ) = h → 0 lim h h ( 4 x + 2 h − 1 )
Cancellation We cancel out h from the numerator and the denominator: f ′ ( x ) = h → 0 lim ( 4 x + 2 h − 1 )
Evaluation Finally, we evaluate the limit as h approaches 0: f ′ ( x ) = 4 x + 2 ( 0 ) − 1 f ′ ( x ) = 4 x − 1
Final Answer Therefore, the derivative of 2 x 2 − x with respect to x from first principle is: 4 x − 1
Examples
Understanding derivatives from first principles is fundamental in physics. For example, if x represents time and f ( x ) = 2 x 2 − x represents the position of an object, then the derivative f ′ ( x ) = 4 x − 1 gives the object's velocity at any time x . This concept is crucial for analyzing motion and predicting the behavior of objects in various physical systems.
