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Basic Differentiation Rules

Objectives

  • State the constant, constant multiple, and power rules.
  • Apply the sum and difference rules to combine derivatives.
  • Use the product rule for finding the derivative of a product of functions.
  • Use the quotient rule for finding the derivative of a quotient of functions.
  • Extend the power rule to functions with negative exponents.
  • Combine the differentiation rules to find the derivative of a polynomial or rational function.

Summary

Given a differentiable function y = f ( x ) , we can express the derivative of f in several different notations: f ′ ( x ) , d f d x , d y d x , and d d x [ f ( x ) ] .

The limit definition of the derivative leads to patterns among certain families of functions that enable us to compute derivative formulas without resorting directly to the limit definition. For example, if f is a power function of the form f ( x ) = x n , then f ′ ( x ) = n x n − 1 for any real number n other than 0. This is called the Rule for Power Functions.

If we are given a constant multiple of a function whose derivative we know, or a sum of functions whose derivatives we know, the Constant Multiple and Sum Rules make it straightforward to compute the derivative of the overall function. More formally, if f ( x ) and g ( x ) are differentiable with derivatives f ′ ( x ) and g ′ ( x ) and a and b are constants, then

d d x [ a f ( x ) + b g ( x ) ] = a f ′ ( x ) + b g ′ ( x ) .

By carefully analyzing the graphs of y = sin ⁡ ( x ) and y = cos ⁡ ( x ) , and by using the limit definition of the derivative at select points, we found that d d x [ sin ⁡ ( x ) ] = cos ⁡ ( x ) and d d x [ cos ⁡ ( x ) ] = − sin ⁡ ( x ) .

We note that all previously encountered derivative rules still hold, but now may also be applied to functions involving the sine and cosine. All of the established meaning of the derivative applies to these trigonometric functions as well.

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