We compute the derivative of a product.

The Product Rule

In words, the derivative of a product is the derivative of the first times the second plus the first times the derivative of the second.

(problem 1) Compute
Use the Product Rule with and .
.
The derivative of is
The derivative of is

The derivative of with respect to is

(problem 2) Compute
Use the Product Rule with and .
.
The derivative of is
The derivative of is

The derivative of with respect to is

Here is a video of the preceding example
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(problem 3a) Compute
Use the Product Rule with and .
.
The derivative of is
The derivative of is

The derivative of with respect to is

(problem 3b) Compute
Use the Product Rule with and .
.
The derivative of is
The derivative of is

The derivative of with respect to is

Here is a video of the preceding example
_
(problem 4a) Compute
Use the Product Rule with and .
.
The derivative of is
The derivative of is

The derivative of with respect to is

(problem 4b) Compute
Use the Product Rule with and .
.
The derivative of is
The derivative of is

The derivative of with respect to is

Here is a video of the preceding example
_
(problem 5) Compute
Use the Product Rule with and .
.
The derivative of is
The derivative of is itself
Factor out and combine like terms
The derivative of with respect to is
Here is a video of the preceding example
_
(problem 6) Compute
Use the Product Rule with and .
.
The derivative of is
The derivative of is
The derivative of with respect to is
Here is a video of the preceding example
_
(problem 7) Compute
Use the Product Rule with and .
.
The derivative of is itself
The derivative of is
Factor out
The derivative of with respect to is
Here is a detailed, lecture style video on the Product Rule:
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We conclude this section with a derivation of the product rule using the definition of the derivative. \begin{align*} \left [f(x)g(x)\right ]' &= \lim _{h \to 0} \frac{f(x+h)g(x+h)-f(x)g(x)}{h}\\[5pt] &= \lim _{h \to 0} \frac{f(x+h)g(x+h)- f(x)g(x+h) + f(x)g(x+h) - f(x)g(x)}{h}\\[5pt] &= \lim _{h \to 0} \frac{f(x+h)g(x+h)- f(x)g(x+h)}{h} + \frac{f(x)g(x+h) - f(x)g(x)}{h} \\[5pt] &= \lim _{h \to 0} \frac{g(x+h)\left [f(x+h)- f(x)\right ]}{h} + \frac{f(x)\left [g(x+h) - g(x)\right ]}{h}\\[5pt] &= g(x)f'(x) + f(x)g'(x). \end{align*}

2024-09-27 13:55:03