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Rules for Differentiating Univariate Functions

Learn the specific rules for differentiating continuous and smooth univariate functions, including constant, power, sum, product, quotient, chain, natural logarithmic, and natural exponential functions.

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Rules for Differentiating Univariate Functions

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  1. Rules for Differentiating Univariate Functions Given a univariate function that is both continuous and smooth throughout, it is possible to determine its derivative by applying one or more of the specific rules of differentiation outlined in the following.

  2. Rules for Differentiating Univariate Functions • Constant Function Rule: The derivative of a constant function is zero. y = f(x) = c where c is a constant.

  3. Rules for Differentiating Univariate Functions • Constant Multiplied by a Function Rule: Let y be equal to the product of a constant c and some function f(x), such that y = cf(x) then

  4. Rules for Differentiating Univariate Functions • Power Rule: Let y = f(x) = xn, where the dependant variable x is raised to a constant value, the power n, then

  5. Rules for Differentiating Univariate Functions • Sum (Difference) Rule: Let y be the sum (difference) of two functions f(x)andg(x). y = f(x) + g(x) then

  6. Rules for Differentiating Univariate Functions • Product Rule: Let y = f(x).g(x), where f(x) and g(x) are two functions of the variable x. Then

  7. Rules for Differentiating Univariate Functions • Quotient Rule: Let y = f(x)/g(x), where f(x) and g(x) are two functions of the variable x and g(x)≠ 0. Then

  8. Rules for Differentiating Univariate Functions • Chain Rule: Let y = f(z), which is a function of another function, z = g(x). Then

  9. Rules for Differentiating Univariate Functions • Natural Logarithmic Rule: Let y = lnf(x), where y is a natural logarithmic function of x. Then

  10. Rules for Differentiating Univariate Functions • Natural Exponential Function Rule: The natural exponential function rule is used when the natural base, e, is raised to a power that is some function of the independent variable x, such as y = ef(x). Then

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