MORE ON DIFFERENTIATION

Differentiation

Taking limits to find the derivative of a function can be very tedious and complicated. The formulas listed below will make differentiating much easier. Each formula is expressed in the regular notation as well as Leibniz notation.

Constant Rule: If f is a constant function, where f(x) = c, then

Power Rule: If f is a power function, where f(x) = xⁿ and n is a real number, then

Note: The graph of the derivative of a power function will be one degree lower than the graph of the original function.

Note: For an example of a power function question, see Example #6 below.

Constant Multiple Rule: If f is a differentiable function and c is a constant, then

Sum Rule: If f and g are differentiable functions, then

In Leibniz notation,

Note: For an example of the sum rule, see Example #7 below.

Difference Rule: If f and g are differentiable functions, then

In Leibniz notation,

Note: For an example of the difference rule, see Example #8 below.

Product Rule: If f and g are differentiable functions, then

In Leibniz notation,

Note: For an example of the product rule, see Example #9 below.

Quotient Rule: If f and g are differentiable functions, then

In Leibniz notation,

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The Chain Rule

The chain rule is used to find the derivatives of compositions of functions. A composite function is a function that is composed of two other functions. For the two functions f and g, the composite function or the composition of f and g, is defined by

The function g(x) is substituted for x into the function f(x). For example, the function F(x)=(2x+6)⁴ could be considered as a composition of the functions, f(x)=x⁴ and g(x)=2x+6. However, it could also be written as a composition of f(x)=(2x)⁴ and g(x)=x+3. Often, a function can be written as a composition of several different combinations of functions.

The chain rule allows us to find the derivative of composite functions. The chain rule states that if f and g are differentiable functions and F(x)=f(g(x)), then F is differentiable and the derivative of F is given by

In Leibniz notation, if y=f(u), u=g(x) and y and u are differentiable functions, then

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