At the beginning of the week, we continued working with and practicing differentiation with the chain rule. We also learned how to anti-differentiate using U-Substitution. The video that we watched before the U-Substitution lesson helped a TON. I think it was very beneficial to have some background knowledge of the process before going into the lesson.
On Thursday and Friday we focused on implicit differentiation. This helped to solidify my understanding of previous concepts we have covered, as well as learn a new one. When we want to find the derivative with respect to y of an equation such as x^2 + y^2 = 25, we must first differentiate both sides of the equation with respect to x. This involves finding the derivative of every part of the function, even the y's, for which we write dy/dx as their derivative. (In this case, the derivative of the y^2 term would be 2y (dy/dx)). This helped me to better understand that we always need to multiply by the derivative of the inside function. When we worked with x's in previous lessons, it did not appear that we multiplied by dx/dx, but in fact we did because multiplying by the derivative of the inside (dx/dx) is the same as multiplying by 1. This is why when we use the chain rule for simple terms like x^2, we only need to multiply by the power and subtract one from the power to get 2x. We can skip the step of multiplying by the derivative of the inside in this case because 2x * (dx/dx) = 2x.
As derivatives are getting harder, I have found that knowing the specific steps it takes to solve a problem has helped me a lot. When I am first learning and practicing a new concept, it is very easy to look back to the steps and see what I need to do next if I ever get stuck.
Steps and Examples of U-Substitution and Implicit Differentiation:
On Thursday and Friday we focused on implicit differentiation. This helped to solidify my understanding of previous concepts we have covered, as well as learn a new one. When we want to find the derivative with respect to y of an equation such as x^2 + y^2 = 25, we must first differentiate both sides of the equation with respect to x. This involves finding the derivative of every part of the function, even the y's, for which we write dy/dx as their derivative. (In this case, the derivative of the y^2 term would be 2y (dy/dx)). This helped me to better understand that we always need to multiply by the derivative of the inside function. When we worked with x's in previous lessons, it did not appear that we multiplied by dx/dx, but in fact we did because multiplying by the derivative of the inside (dx/dx) is the same as multiplying by 1. This is why when we use the chain rule for simple terms like x^2, we only need to multiply by the power and subtract one from the power to get 2x. We can skip the step of multiplying by the derivative of the inside in this case because 2x * (dx/dx) = 2x.
As derivatives are getting harder, I have found that knowing the specific steps it takes to solve a problem has helped me a lot. When I am first learning and practicing a new concept, it is very easy to look back to the steps and see what I need to do next if I ever get stuck.
Steps and Examples of U-Substitution and Implicit Differentiation: