Product rule for three or more functions

 
 
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How to expand the product rule from two to three functions

Product rule is a derivative rule that allows us to take the derivative of a function which is itself the product of two other functions. Product rule tells us that the derivative of an equation like

???y=f(x)g(x)???

will look like this:

???\frac{dy}{dx}=f\prime(x)g(x)+f(x)g\prime(x)???

But what do we do if our function is the product of more than two functions? In this situation, if our function is

???y=f(x)g(x)h(x)???

then the derivative looks like this:

???\frac{dy}{dx}=f\prime(x)g(x)h(x)+f(x)g\prime(x)h(x)+f(x)g(x)h\prime(x)???

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We can see that the original function was a product of three functions, and its derivative was the sum of three products. If our function was the product of four functions, the derivative would be the sum of four products.

As you can see, when we take the derivative using product rule, we take the derivative of one function at a time, multiplying by the other two original functions. To be more specific, we take the derivative of ???f(x)???, and multiply it by ???g(x)??? and ???h(x)???, leaving those two as they are. Then we add to that the derivative of ???g(x)???, multiplied by ???f(x)??? and ???h(x)??? left as they are. We can continue this pattern, taking the derivative of only one of the functions and leaving the others alone, for as many functions as are multiplied together in our original problem.

Looking at an equation which is the product of four functions, like

???y=f(x)g(x)h(x)j(x)???

the derivative is

???\frac{dy}{dx}=f\prime(x)g(x)h(x)j(x)+f(x)g\prime(x)h(x)j(x)+f(x)g(x)h\prime(x)j(x)+f(x)g(x)h(x)j\prime(x)???

 
 

Video example of applying the product rule for derivatives to the product of three functions


 
 
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Product rule for the product of a power, trig, and exponential function

Example

Use product rule to find the derivative.

???y=\left(4x^6\right)\left(\sin{3x}\right)\left(-6e^{2x}\right)???

Applying product rule, we get

???\frac{dy}{dx}=\left(24x^5\right)\left(\sin{3x}\right)\left(-6e^{2x}\right)+\left(4x^6\right)\left(3\cos{3x}\right)\left(-6e^{2x}\right)+\left(4x^6\right)\left(\sin{3x}\right)\left(-12e^{2x}\right)???

???\frac{dy}{dx}=-144x^5e^{2x}\sin{3x}-72x^6e^{2x}\cos{3x}-48x^6e^{2x}\sin{3x}???

???\frac{dy}{dx}=-24x^5e^{2x}(6\sin{3x}+3x\cos{3x}+2x\sin{3x})???

product rule for derivatives with more than two functions.jpg

To apply product rule to three or more functions, include a term that includes each combination of the derivative of one function, And where all other functions are held constant


Let’s do another example with more functions.


Example

Use product rule to find the derivative.

???y=\left(8x^{12}\right)\left(\frac{6x^2}{7}\right)\left(-2e^x\right)\left(-6\cos{5x}\right)\left(\sin{4x}\right)???

Applying product rule, we get

???\frac{dy}{dx}=\left(96x^{11}\right)\left(\frac{6x^2}{7}\right)\left(-2e^x\right)\left(-6\cos{5x}\right)\left(\sin{4x}\right)???

???+\left(8x^{12}\right)\left(\frac{12x}{7}\right)\left(-2e^x\right)\left(-6\cos{5x}\right)\left(\sin{4x}\right)???

???+\left(8x^{12}\right)\left(\frac{6x^2}{7}\right)\left(-2e^x\right)\left(-6\cos{5x}\right)\left(\sin{4x}\right)???

???+\left(8x^{12}\right)\left(\frac{6x^2}{7}\right)\left(-2e^x\right)\left(30\sin{5x}\right)\left(\sin{4x}\right)???

???+\left(8x^{12}\right)\left(\frac{6x^2}{7}\right)\left(-2e^x\right)\left(-6\cos{5x}\right)\left(4\cos{4x}\right)???

Pulling out common factors, we get

???\frac{dy}{dx}=\left[\left(8x^{11}\right)\left(\frac{6x}{7}\right)\left(-2e^x\right)\left(-6\right)\right]\cdot\Big[\left(12\right)\left(x\right)\left(\cos{5x}\right)\left(\sin{4x}\right)???

???+(x)(2)\left(\cos{5x}\right)\left(\sin{4x}\right)???

???+(x)(x)\left(\cos{5x}\right)\left(\sin{4x}\right)???

???+(x)(x)\left(-5\sin{5x}\right)\left(\sin{4x}\right)???

???+(x)(x)\left(\cos{5x}\right)\left(4\cos{4x}\right)\Big]???

And simplifying, we get

???\frac{dy}{dx}=\left(\frac{576}{7}e^xx^{12}\right)\cdot\Big[12x\sin{4x}\cos{5x}???

???+2x\sin{4x}\cos{5x}???

???+x^2\sin{4x}\cos{5x}???

???-5x^2\sin{4x}\sin{5x}???

???+4x^2\cos{4x}\cos{5x}\Big]???

???\frac{dy}{dx}=\left(\frac{576}{7}e^xx^{13}\right)\cdot\Big[12\sin{4x}\cos{5x}???

???+2\sin{4x}\cos{5x}???

???+x\sin{4x}\cos{5x}???

???-5x\sin{4x}\sin{5x}???

???+4x\cos{4x}\cos{5x}\Big]???

 
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