This week we explored three major topics: using the chain rule to identify the integral, implicit differentiation, and higher order derivatives of implicit functions.
I found the video notes to be extremely helpful for the activity we did Monday. For the most-part, I understand how to find the integral. I think using (u) as opposed to reasoning my way through the problem is much more efficient! The only thing that I struggled with was understanding that when finding the anti-derivative of a trig. function, I did not have to use the power rule like I had to with other functions. Once I understood this, the problems became much easier.
Essentially, to find the integral of the problem: (x^2 + 3)^3 2x dx
You would do the following:
Assign a (u): (u)^3 2x dx, u= x^2 +3
Identify (du): (du)/(dx)=2x -------> (du)=2xdx
Substitute (du): (u)^3 (du)
Identify (f(u)): f(u)=(u)^4/(4) + c *DON'T FORGET THE C*
Substitute the (x): f(x)=(x^2 + 3)^4/(4) +c
After grasping this we moved on to implicit differentiation. I thought this wasn't too challenging, just time consuming sometimes. One thing I struggled with at the beginning was remembering that (Y) IS A FUNCTION, so the derivative of (y) always has to be multiplied with (dy/dx). It gets sort of messy too when we start looking at implicit functions in which we have to use the quotient or product rule.
Basically there are four steps to finding the derivative of implicit differentiation problems:
1. Differentiate both sides with respect to (x)
2. Collect terms with (dy/dx) on one side
3. Factor out (dy/dx)
4. Solve for (dy/dx)
Here are a few examples of implicit differentiation problems:
I found the video notes to be extremely helpful for the activity we did Monday. For the most-part, I understand how to find the integral. I think using (u) as opposed to reasoning my way through the problem is much more efficient! The only thing that I struggled with was understanding that when finding the anti-derivative of a trig. function, I did not have to use the power rule like I had to with other functions. Once I understood this, the problems became much easier.
Essentially, to find the integral of the problem: (x^2 + 3)^3 2x dx
You would do the following:
Assign a (u): (u)^3 2x dx, u= x^2 +3
Identify (du): (du)/(dx)=2x -------> (du)=2xdx
Substitute (du): (u)^3 (du)
Identify (f(u)): f(u)=(u)^4/(4) + c *DON'T FORGET THE C*
Substitute the (x): f(x)=(x^2 + 3)^4/(4) +c
After grasping this we moved on to implicit differentiation. I thought this wasn't too challenging, just time consuming sometimes. One thing I struggled with at the beginning was remembering that (Y) IS A FUNCTION, so the derivative of (y) always has to be multiplied with (dy/dx). It gets sort of messy too when we start looking at implicit functions in which we have to use the quotient or product rule.
Basically there are four steps to finding the derivative of implicit differentiation problems:
1. Differentiate both sides with respect to (x)
2. Collect terms with (dy/dx) on one side
3. Factor out (dy/dx)
4. Solve for (dy/dx)
Here are a few examples of implicit differentiation problems:
The last thing we talked about was finding the higher order derivatives of implicit differentiation functions. This was pretty simple seeing as we already know how to find the first derivative of implicit functions and we already know how to find the second, third, fourth, etc. derivatives of non-implicit functions. I think the main thing that gets me stuck sometimes is remembering that when I am finding the second derivative of an implicit differentiation function, I have to multiply by (d/dx).