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Mathematical Expression Editor
The following rules allow us the find the derivative of multiples, sums and differences
of functions whose derivatives are already known.
_
1 Constant Multiple Rule
Constant Multiple Rule If is differentiable and is any constant, then
In words, the derivative of a constant times a function is the constant times the
derivative of the function.
example 1 If then we use the constant multiple rule with and we get
(problem 1a) Compute
The derivative of is
The derivative of with respect to is
(problem 1b) Compute
Use the Power Rule on
The Power Rule says:
Don’t forget to multiply by
The derivative of with respect to is
example 2 If then we use the constant multiple rule with and we get
(problem 2a) Compute
The derivative of is
The derivative of with respect to is
(problem 2b) Compute
Use the Power Rule on
The Power Rule says:
Don’t forget to multiply by
The derivative of with respect to is
(problem 2c) Compute
Use the Power Rule on
The Power Rule says:
Don’t forget to multiply by
The derivative of with respect to is
example 3 If then we rewrite as and we use the constant multiple rule with , giving
(problem 3a) Compute
Rewrite using
Use the Power Rule on
The Power Rule says:
Don’t forget to multiply by 10
The derivative of with respect to is
(problem 3b) Compute
Rewrite using
Use the Power Rule on
The Power Rule says:
Don’t forget to multiply by 6
The derivative of with respect to is
2 Sum Rule
Sum Rule If and are differentiable functions then
In words, the derivative of a sum is the sum of the derivatives.
example 4 If then .
(problem 4a) Compute
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
(problem 4b) Compute
The Sum Rule can be used on more than two terms
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
example 5 If then we rewrite as and
(problem 5a) Compute
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
(problem 5b) Compute
The Sum Rule can be used on more than two terms
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
example 6 If then we rewrite as and \begin{align*} f'(x) &= \tfrac 12 x^{-1/2} + \tfrac 13 x^{-2/3} \\ &= \tfrac 12 \cdot \frac{1}{x^{1/2}} + \tfrac 13 \cdot \frac{1}{x^{2/3}}\\ &= \frac{1}{2\sqrt x} + \frac{1}{3\sqrt [3]{x^2}}. \end{align*}
(problem 6a) Compute
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
(problem 6b) Compute
The Sum Rule can be used on more than two terms
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
3 Difference Rule
Difference Rule If and are differentiable functions, then
In words, the derivative of a difference is the difference of the derivatives.
example 7 If then .
(problem 7a) Compute
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
(problem 7b) Compute
The Difference Rule can be used on more than two terms
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
(problem 7c) Compute
The Difference Rule can be used on more than two terms
The Sum, Difference and Constant Multiple Rules are often used together
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
example 8 If then we rewrite as and .
(problem 8a) Compute
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
(problem 8b) Compute
The Difference Rule can be used on more than two terms
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
example 9 If then we rewrite as and .
(problem 9a) Compute
Use the Power Rule on each term
The Power Rule says:
The derivative of with respect to is
(problem 9b) Compute
The Sum and Difference Rules can be used on more than two terms