product rule
The product rule is a fundamental rule in calculus used to find the derivative of two functions that are multiplied together
The product rule is a fundamental rule in calculus used to find the derivative of two functions that are multiplied together.
If we have two functions, f(x) and g(x), the product rule states that the derivative of their product, f(x) * g(x), is given by the following formula:
(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)
In other words, to differentiate the product of two functions, we differentiate one function while keeping the other unchanged, then add it to the differentiation of the other function while keeping the first one unchanged.
Let’s look at an example to understand how to apply the product rule. Suppose we want to find the derivative of the function h(x) = x * sin(x). Using the product rule, we can differentiate this function as follows:
h'(x) = (x)’ * sin(x) + x * (sin(x))’
Using the derivatives of x and sin(x), which are 1 and cos(x) respectively, we can simplify this expression:
h'(x) = 1 * sin(x) + x * cos(x) = sin(x) + x * cos(x)
Therefore, the derivative of h(x) = x * sin(x) is h'(x) = sin(x) + x * cos(x).
The product rule is a powerful tool that allows us to find the derivative of functions that involve multiplication, which is particularly useful in various branches of mathematics and science, such as physics and engineering.
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