Let . Note that we can write as a composition of functions, , where , , and .

To compute , we use the chain rule:

For this, we first need the derivatives of , and :

, so .

, so .

, since .

Combining these three results, we get:

.

Problem 2 asks for the equation of the line tangent to at .

For this, we first find a point on the line, and its slope. Since we are told that the line is tangent to the given curve when , this means that the line goes through the point , where the value of is found by replacing in the given function. This gives us

To find the slope of the tangent line, we recall that this is just the value of the derivative of the given function, when . The derivative is given by

Again, this is when .

This means that the tangent line we require goes through the point and has slope , i.e., it is simply the line .

Here is the graph of the function , with the tangent line and the line indicated.

Click the image above to enlarge.

43.614000-116.202000

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