The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Derivative of a natural log: 

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function  at the point 

The slope of the tangent is

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Derivative of a natural log: 

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function

The slope of the tangent is  at the point 

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Derivative of a natural log: 

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function  at the point 

The slope of the tangent is

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Derivative of an exponential: 

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function  at the point 

The slope of the tangent is

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Derivative of an exponential: 

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function  at the point 

The slope of the tangent is

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Derivative of an exponential: 

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function  at the point 

The slope of the tangent is

The first derivative of the given function makes use of the power rule of derivatives. 

If you have a function

According to the Power Rule, the first derivative is defined as 

If this rule is applied to each of the terms in the function f(x) we get that 

To find the first derivative of this particular function is accomplished by applying the power rule which states,

Applying the above rule to the equation,

results in,

To take the derivative, you first use the power rule for differentiating, 

,

then you use the chain rule, 

.

Also recall the trigonometric rule for differentiating sine,

This produces,

. 

Use the quotient rule to find the derivative which states,

.

Given,

the derivatives can be found using the power rule which states,

therefore,

Applying the quotient rule to our function we find the derivative to be as follows.

Simplify.