Definition from Limits
The derivative is defined as a limit: the limit of the difference quotient as the interval shrinks to zero.
The Power Rule
The Product Rule
The Quotient Rule
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Tangent Lines and Geometric Interpretation
Optimization: Finding Maxima and Minima
Common Mistakes
Forgetting the chain rule
When differentiating composite functions like sin(3x), students often forget to multiply by the derivative of the inner function (3). The result should be 3cos(3x), not cos(3x).
Mixing up the product and quotient rules
The product rule adds two terms: f'g + fg'. The quotient rule subtracts: (f'g - fg')/g². Confusing the sign or the order leads to incorrect results.
Practice Problems
Related Guides
Frequently Asked Questions
What is a derivative?
A derivative measures the instantaneous rate of change of a function. It tells you how fast the output changes when the input changes by a tiny amount. Geometrically, the derivative at a point is the slope of the tangent line to the graph at that point.
What is the power rule?
The power rule states that the derivative of x^n is n·x^(n-1). For example, the derivative of x³ is 3x², and the derivative of x^(-2) is -2x^(-3). It works for any real exponent, including fractions and negatives.
What is the chain rule?
The chain rule is used to differentiate composite functions: the derivative of f(g(x)) is f'(g(x)) · g'(x). In words, differentiate the outer function (evaluated at the inner function) and multiply by the derivative of the inner function.
When do you use the product rule vs the chain rule?
Use the product rule when two functions are multiplied together: (fg)' = f'g + fg'. Use the chain rule when one function is inside another (composition): [f(g(x))]' = f'(g(x))·g'(x). They address fundamentally different situations.
What does a derivative of zero mean?
A derivative of zero at a point means the function has a horizontal tangent line there — it is momentarily not increasing or decreasing. This point is a critical point and could be a local maximum, local minimum, or inflection point. Further analysis (second derivative test) determines which.
How are derivatives used in real life?
Derivatives are used to find rates of change (velocity from position, acceleration from velocity), optimize functions (maximize profit, minimize cost), analyze function behavior (increasing/decreasing intervals, concavity), and model dynamic systems in physics, engineering, economics, and biology.
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