Definition of the Derivative

TL;DR

Learn how to find the derivative of a function using the limit process and the formula f prime of x = (f(x+h) - f(x))/h.

Transcript

in this video we're going to talk about how to find the derivative using the definition of the derivative formula so basically we need to find the derivative of a function using the limit process and f prime of x represents the derivative of f of x and it's equal to the limit as h approaches zero of f of x plus h minus f of x divided by h so that's... Read More

Key Insights

• 😑 The derivative of a linear function can be found by substituting values into the derivative formula and simplifying the expression.
• ❓ The derivative of a quadratic function can also be found using the same process, resulting in a linear function.
• 😑 To find the derivative of functions with radicals or fractions, multiply the numerator and denominator by the conjugate to eliminate radicals and simplify the expression.

Questions & Answers

Q: How do you find the derivative of a linear function?

To find the derivative of a linear function, substitute the function into the derivative formula f prime of x = (f(x+h) - f(x))/h and simplify the expression.

Q: What is the derivative of the function x^2?

To find the derivative of x^2, substitute the function into the derivative formula and simplify. The derivative is 2x.

Q: How do you find the derivative of 1/x?

To find the derivative of 1/x, substitute the function into the derivative formula, multiply the numerator and denominator by the conjugate, and simplify. The derivative is -1/x^2.

Q: What is the derivative of the square root of x?

To find the derivative of the square root of x, substitute the function into the derivative formula, multiply the numerator and denominator by the conjugate, and simplify. The derivative is 1/(2sqrt(x)).

Q: How do you find the derivative of a polynomial function?

To find the derivative of a polynomial function, substitute the function into the derivative formula, expand and distribute terms, cancel out like terms, factor out the common factors, and simplify.

Q: What is the derivative of x^2 - 5x + 9?

To find the derivative of x^2 - 5x + 9, substitute the function into the derivative formula, simplify the expression, and cancel out terms. The derivative is 2x - 5.

Summary & Key Takeaways

• The video explains how to find the derivative of a linear function by substituting values into the derivative formula and simplifying the expression. The derivative of 5x - 4 is 5.

• The video then demonstrates finding the derivative of the function x^2 using the same process. The derivative of x^2 is 2x.

• The video also shows how to find the derivative of the function 1/x and the square root of x using the definition formula. The derivatives are -1/x^2 and 1/(2sqrt(x)).

• Finally, the video explains how to find the derivative of a polynomial function, using the example of x^2 - 5x + 9. The derivative is 2x - 5.