Using the Product Rule

The Product Rule is for finding the derivative of a function that consists of the product of two functions.

If we have two functions multiplied together:

y = u * v

The derivative of this product of two functions is:

dy/dx = u * dv/dx + v * du/dx

Like this: y = x^{3} (x^{5 }+ 3)

Let u = x^{3} and v = x^{5} + 3.

Then: dy/dx = x^{3} * (5x^{4}) + (x^{5} +3) * 3x^{2}

dy/dx = 5x^{7} + 3x^{7} + 9x^{2}

dy/dx = 8x^{7} +9x^{2}

(Of course, an easier way to solve this is to multiply it out before you take the derivative, but we wouldn