Find the line tangent to  at .

First, we find :

Next, we find the derivative:

Therefore, the slope at  is:

.

Using point-slope form, we can write the tangent line:

Simplifying this gives us:

The other point of the triangle must be at  as it must be equidistant from the other two points of the triangle. Since all points on the y-axis are  units away from the other points in the  direction, the third point must be equidistant in the  direction from both  and . The distance between these points is , so the third point must have a y-value of . The third point is now at  so the slope of the line from  to  is as follows.

We must take the derivative of the function using the chain rule yielding .

The chain rule is .

Also remember that the derivative of  is .

Applying these rules we get the following.

Plugging in the value for  we get  which is .

A line is tangent to a point on a curve when its slope is equivalent to the slope of the curve at the point of intersection. Therefore to solve this equation, the slope of the curve at  must be found.  

The slope of a graph at any point can be found by taking the first derivative.  To take the derivative of this equation, we must use the power rule,  

.  

We also must remember that the derivative of an constant is 0.  By taking the first derivative of the graph equation, we obtain the slope equation 

.  

Plugging in , we find the slope at that point is 13, therefore any line tangent to the curve at that point must have a slope of 13.

To find the a general formula for the slope of the function , derive the function with respect to :

The slope of the function at  is then found as:

The slope of a tangent line to a function at a point can be found by taking the derivative of the function and plugging in the point at which the slope is to be found. The derivative of the funtion can be found using the product rule.

The derivative of  is .

Also, the derivative of a sin function is the cos function.

The slope of the tangent at a certain x is the value of the derivative at that point. The derivative of  is .

The derivative of the given function is 

.

Plugging in x=3 gives

.

The derivative of a function describes the slope. Therefore, you must first find the derivative of the function, which is .

Then, you plug in the specific x value given in the problem, which is -1:

.

Therefore, your final answer is -7.

First, you must find the slope equation of the tangent line to the function, which is just the derivative of the function:

.

Since that is the slope equation, you need to set that equal to 0 and factor:

, or , which yields .

If the two lines are perpendicular, then their slopes must be opposite reciprocals. Since both lines are tangent to the curve, , their slopes will be equal to the slope of the curve at the points to which they are tangent. So  has slope  and  has slope . Following this, if the lines are perpendicular, then .