00:01
In this question, we are asked to find an equation of the tangent line to the given curve at the given point.
00:06
And to do that, we need to differentiate.
00:09
We will first assume that y is a function which depends on x, and x is the independent parable, and then differentiate both sides of the given equation with respect to x.
00:40
What we're going to do next is just differentiate.
00:44
So the derivative of 1 is 0.
00:48
Next we need to calculate the derivative of ln of 3xy.
00:53
By the chain rule, we need to differentiate the expression inside the logarithm, d over d x of 3xy, and multiply this by the derivative of the logarithm, which is 1 over 3xy.
01:11
On the right hand side, we need to differentiate the power of the exponential function and multiply this by the derivative of the exponential function, which is the exponential function itself.
01:33
Now, the derivative of 3xy by the product rule equals to 3 times the derivative of x multiplied by y plus 3x multiplied by the derivative of y.
01:55
And all of this gets multiplied by 1 over 3xy.
02:01
And on the right hand side, we're going to get 3 times the derivative of x minus the derivative of y times e to the 3x minus y.
02:23
Now, the derivative of x equals to 1, right? because x is the independent variable.
02:30
And the derivative of y with respect to x is just dy over the x.
02:36
We call it that we can't just write down 1 because y depends on x.
02:41
So we are going to get 3y plus 3x times the y over d x times 1 over 3xy equals to 3x sorry 3 the derivative of x is 1 minus d y times e to the 3x minus y.
03:19
Now we need to solve this for the y over d x.
03:24
First of all we can cancel 3.
03:28
Then we are going to get after distributing we are going to get y over xy plus x over xy times dy over d x equals and on the right hand side we are going to get three times e to the 3 x minus y minus e times d y times d y over d x now on the left hand side we can cancel y in the first fraction and we can cancel x in the second fraction now we can cancel x in the second fraction now we let's move the terms with dy over dx to the left -hand side and everything else to the right -hand side.
04:16
On the left -hand side, we are going to get 1 over y times d -y over dx plus e to 3x minus y times d -y equals to 3e to the 3x minus y minus 1 over x...