Just started derivatives in Math 262 and we are learning the chain rule.
Then I get this:
f(x) = 2x + (2x + (2x +1)^3)^3
Her is what I'm attempting...
2x + 3(2x + 3(2x + 1)^2)^2
Is this even close?
Thanks for any help you might have.
Just started derivatives in Math 262 and we are learning the chain rule.
Then I get this:
f(x) = 2x + (2x + (2x +1)^3)^3
Her is what I'm attempting...
2x + 3(2x + 3(2x + 1)^2)^2
Is this even close?
Thanks for any help you might have.
First of all, the derivative of a sum is equal to the sum of the derivatives. So the derivative of the first 2x is 2. The hard part will be evaluating the derivative of $\displaystyle \displaystyle \begin{align*} \left[ 2x + \left( 2x + 1 \right) \right]^3 \end{align*}$. It's a composition of functions, so the chain rule will need to be used. I always use Leibnitz notation for the chain rule, since it is easier. In this case you will need to do $\displaystyle \displaystyle \begin{align*} \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \end{align*}$.
So first, if we have $\displaystyle \displaystyle \begin{align*} y = \left[ 2x + \left( 2x + 1 \right)^3 \right]^3 \end{align*}$, then we let $\displaystyle \displaystyle \begin{align*} u = 2x + \left( 2x + 1 \right)^3 \implies y = u^3 \end{align*}$. Then $\displaystyle \displaystyle \begin{align*}\frac{dy}{du} = 3u^2 = 3 \left[ 2x + \left( 2x + 1 \right)^3 \right]^2 \end{align*}$.
As for finding $\displaystyle \displaystyle \begin{align*} \frac{du}{dx} \end{align*}$, we notice that $\displaystyle \displaystyle \begin{align*} u = 2x + \left( 2x + 1 \right)^3 \implies \frac{du}{dx} = 2 + \frac{d}{dx} \left[ \left( 2x + 1 \right)^3 \right] \end{align*}$. Can you use the Chain Rule to evaluate this final derivative?
When you evalute $\displaystyle \displaystyle \begin{align*} \frac{d}{dx}\left[ \left(2x+ 1 \right)^3 \right] \end{align*}$, the "inner" function is u = 2x + 1. What's its derivative?
The "outer" function is $\displaystyle \displaystyle \begin{align*} u^3 \end{align*}$. What's its derivative?
What do you get when you multiply them together?