At the beginning of the week we went a little further with chain rule. Basically it's taking the anti-derivative of a function using the chain rule. We did this by doing 'U' substitution. The steps to solving are as follows:
1. Find u
2. Find u' (du/dx)
3. Solve for du
4. Write function in terms of u
5. Anti-deriv the function (Add 1 to the exponent and divide)
6. Substitute u back in and add c
An example of this would be if we were taking the anti-derivative of 4sin(5-4x)dx. The first step would be to define u. In this case, u would be (5-4x). Then take the derivative of u so du/dx=-4. Solve for du by multiplying both sides by dx so you get du=-4dx. We want to get 4dx, so you would divide by -1 and get -du=4dx. Next, we would write the function in terms of u so it would be -sinudu. f(u)=--cosu, which would equal cosu. Finally, f(x)=cos(5-4x)+c would be your final answer.
The other section we covered this week was on implicit differentiation. The idea of implicit differentiation doesn't seem too bad, but as the problems got harder I struggled with it more. the steps for solving are as follows:
1. Differentiate both sides with repeat to x
2. Collect terms with dy/dx to one side
3. Factor out dy/dx
4. Solve for dy/dx
The most important thing to remember when solving is that y is a function!! An example of this would be x^2-xy+y^2=7. First, I'd differentiate both sides so it would read 2x*dy/dx+y(-1)+2y(dy/dx). Next I'd collect the dy/dx on one side so it would read 2x-y=xdy/dx-2ydy/dx. Then I'd factor out the dy/dx so it would read dy/dx(x-2y)=2x-y. Solve for dy/dx by dividing by (x-2y) so your final answer would be dy/dx=(2x-y)/(x-2y)
1. Find u
2. Find u' (du/dx)
3. Solve for du
4. Write function in terms of u
5. Anti-deriv the function (Add 1 to the exponent and divide)
6. Substitute u back in and add c
An example of this would be if we were taking the anti-derivative of 4sin(5-4x)dx. The first step would be to define u. In this case, u would be (5-4x). Then take the derivative of u so du/dx=-4. Solve for du by multiplying both sides by dx so you get du=-4dx. We want to get 4dx, so you would divide by -1 and get -du=4dx. Next, we would write the function in terms of u so it would be -sinudu. f(u)=--cosu, which would equal cosu. Finally, f(x)=cos(5-4x)+c would be your final answer.
The other section we covered this week was on implicit differentiation. The idea of implicit differentiation doesn't seem too bad, but as the problems got harder I struggled with it more. the steps for solving are as follows:
1. Differentiate both sides with repeat to x
2. Collect terms with dy/dx to one side
3. Factor out dy/dx
4. Solve for dy/dx
The most important thing to remember when solving is that y is a function!! An example of this would be x^2-xy+y^2=7. First, I'd differentiate both sides so it would read 2x*dy/dx+y(-1)+2y(dy/dx). Next I'd collect the dy/dx on one side so it would read 2x-y=xdy/dx-2ydy/dx. Then I'd factor out the dy/dx so it would read dy/dx(x-2y)=2x-y. Solve for dy/dx by dividing by (x-2y) so your final answer would be dy/dx=(2x-y)/(x-2y)