Product Rule | A way of calculating derivatives by multiplying two functions’ derivatives together. Product Rule Each derivative of a function is included in the derivative of the product. A way to calculate the derivative of a product using calculus
A product of two functions, u(x) and v(x), has a derivative equal to the derivative of the first function, u(x), multiplied by the second function, v(x), plus the derivative of the first function, u(x), multiplied by the derivative of the dual function, v(x).
Mathematically, it can be written as:
(d/dx) [u(x) * v(x)] = u(x) * (d/dx) [v(x)] + v(x) * (d/dx) [u(x)]
This rule can be used to find the derivative of a product of two functions by finding the derivatives of the individual functions.
The product rule is a specific case of the chain rule when two functions are multiplied.
The product rule is a handy way of calculating derivatives. By definition, the derivative of the product of two functions is the product of their derivatives.
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This can be written mathematically as:
Product Rule Formula
(f*g)’ = f’*g + f*g’
It’s important to note that the product rule only applies when the functions are being multiplied, not added or subtracted.
Let’s take a look at an example.
Suppose we want to calculate the derivative of x*sin(x).
We can use the product rule to break this down into two simpler problems:
x’*sin(x) and sin(x)*x’.
The derivative of sin(x) is cos(x), so the derivative of x’*sin(x) is just x’. The derivative of sin(x)*x’ is sin(x)*x’ + cos(x)*x’, but we can simplify this to sin(x)*x’ + 1.
So the final answer is:
dx/dX = (sin(x)*x’) + 1
A product rule is a helpful tool for solving derivatives, and it’s worth taking the time to practice using it.
With a bit of practice, you’ll be able to apply it in any situation.
If you’re looking for more practice problems, be sure to check out our quiz on derivatives.
It covers all the, and will help you apply the product rule.
The product rule is a way to calculate derivatives by multiplying two functions’ derivatives together.
It can be written in terms of infinitesimals as the following:
(f*g)’ = f’*g + f*g’
First, separate all the infinitesimal terms on one side of the equation so that there are no infinitesimals on both sides of the equation.
This gives us
Δx · Δy = [Δx]’ * [Δy] + [Δx] * [Δy]’
For differentiation, we can use the product rule, which is a product of two functions multiplied together.
The product rule states that when taking the derivative of a product, we take the derivatives of each function and multiply them together.
This gives us
Δx · Δy = (Δx)’ * (δy) + (Δx)*(δy)’
Lastly, cancel out like terms on each side of the equation.
Δx · Δy = (Δx)’ * (δy)
Product Rule Of Differentiation
The product rule is often used to differentiate functions of two or more functions. product rule differentiation
When using this rule, it is essential to remember that the derivatives of the functions must be multiplied together to find the derivative of the product.