The equation of the tangent line has the form .

The slope  can be determined by evaluating the derivative of the function at .

Plugging this into the point slope equation, we get

 can be determined by evaluating the original function at .

Plugging this into the previous equation and simplifying gives us

To find the slope of the line tangent you must take the derivative of the function. The derivative of cosine is negative sine and the derivative of sine is cosine.

This makes the derivative of the function

.

Plug in the given x to get the slope.

To find the slope of the line tangent to the function at a point you must first find the derivative.

The power rule states that the derivative of  is .

The derivative of  is .

The derivative of the function is

.

Plugging in 1 for x gives 

.

" is decreasing at ." is incorrect.  The function is increasing at because .

" is increasing over the interval ." is possibly true, but there is not enough information to conclude that it must be true.

" has a point of inflection at ." is possibly true.  Although we know that , a requirement for an inflection point, we do not know that changes signs at .

" must have at least one relative maximum." is possibly true, but there is not enough information to conclude that it must be true.

"The line is tangent to ." must be true.  Because , the function travels through the point .  Because , the slope of the line tangent to the curve at is 5.  Use point-slope form to determine the equation of the tangent line.

The derivative of the function is , and is found using the power rule

and the rule for the derivative of natural log which is, 

so plugging in  gives , which must be the slope of the line since the tangent line's slope is determined by the derivative.

Thus, the line is of the form , where b is unknown.

Solve for b by setting the equation equal to  and plugging in  for x since that is the given point. 

, which gives us 

In order to find the slope of the tangent line through a certain point, we must find the rate of change (derivative) of the function.  The derivative of   is written as .  This tells us what the slope of the tangent line is through any point  in our function .  In other words, all we need to do is plug-in  (because our point  has an x-value of 1) into .  This will give us our answer,  .

To find the tangent line one must first find the slope, this can be given by the derivative evaluated at a point.

To find the derivative of this function use the power rule which states,

The derviative of  is .

Evaluated at our point ,

we find that the slope, m is also 3.

Now we may use the point-slope equation of a line to find the tangent line.

The point slope equation is 

 Where  is the point at which the line is tangent.

Using this definition we find the tangent line to be defined by .