y = (g)^5 , so = 5(g)^4 , where g = 5x^3 + 3xThen we multiplied by the derivative of g: (15x^2 + 3).
Always do it this way. The process has several successive steps, like peeling away the layers of an onion
until you reach the center.
Example 8: If y = , then = (x^3 − 4x)
−
(3x^2 − 4)Again, we took the derivative of the outside function, leaving the inside alone. Then we multiplied by the
derivative of the inside.
Example 9: If y = , then
= [(x^5 - 8x^3 )(x^2 + 6x)]−
[(x^5 - 8x^3 )(2x +6)(x^2 + 6x)(5x^4 − 24x^2 )]Messy, isn’t it? That’s because we used the Chain Rule and the Product Rule. Now for one with the Chain
Rule and the Quotient Rule.
Example 10: If y = , then
Example 11: if y = , then = (5x^3 + x)
−
(15x^2 + 1)Now we use the Product Rule and the Chain Rule to find the second derivative.
= (5x^3 + x)−
(30x) + (15x^2 + 1) You can also simplify this further, if necessary.
There’s another representation of the Chain Rule that you need to learn.