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Understanding Derivatives: Definitions, Rules, and Applications

This tutorial from The Math Center provides a comprehensive overview of derivatives. Learn about the definition of a derivative, various interpretations such as the slope of the tangent line and instantaneous rates of change. Explore essential derivative rules like the Power Rule, Constant Rule, Logarithm Rule, and Exponential Rule. The tutorial includes practical examples and solutions to help you understand how to apply these rules to calculate derivatives effectively. Whether you seek to find instantaneous velocity or analyze functions, this resource equips you with vital skills.

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Understanding Derivatives: Definitions, Rules, and Applications

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  1. Basic Derivatives Brought To You By: Tutorial Services The Math Center

  2. Definition Of A Derivative The derivative of a function at a point x=a, can be interpreted in several different ways: • Algebraically, the • Geometrically as the slope of the tangent line to the graph of at • Functionally as the instantaneous rate of change of at

  3. Applications of Derivatives • Finding the instantaneous velocity of an object at a precise moment in time • Finding the instantaneous rate of change of a function • Finding the slope of the tangent to the graph of a function * NOTE: the derivative symbol can be written as or or

  4. Basic Derivative Formulas • If , where n is a constant, then • If , where c is a constant, then • If , where c is a constant, then Power Rule Constant Rule Power Rule

  5. More basic derivative formulas Logarithm Rule If , then Exponential Rule If , then

  6. Examples Solution: Following the Power Rule, we can now calculate the derivative. Differentiate:

  7. Examples (cont.) Differentiate: Solution: Following the Constant Rule,we can now calculate the derivative.

  8. Examples (cont.) Differentiate: Solution: Following the Power Rule, we can now calculate the derivative.

  9. Examples (cont.) Differentiate: Solution: Following the Logarithm Rule, we can now calculate the derivative. u = x u’ = 1

  10. Examples (cont.) Differentiate: Solution: Following the Exponential Rule,we can now calculate the derivative.

  11. Questions???

  12. Helpful Links • Derivatives and Integrals Handout • Implicit Differentiation Handout • Derivatives Student Handout • Derivatives Quiz

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