This function is implicit, because y is not defined directly in terms of only x. We could try to solve for y, but that would be difficult, if not impossible. The easier solution would be to employ implicit differentiation. Our strategy will be to differentiate the left and right sides by x, apply the rules of differentiation (such as Chain and Product Rules), group dy/dx terms, and solve for dy/dx in terms of both x and y.

We will need to apply both the Product Rule and Chain Rule to both the xlny and the  terms.

According to the Product Rule, if f(x) and g(x) are functions, then .

And according to the Chain Rule,

Now we will group the dy/dx terms and move everything else to the opposite side.

Then, we can solve for dy/dx.

To remove the compound fraction, we can multiply the top and bottom of the fraction by y.

The (ugly) answer is .

This derivative uses the power rule. Keep in mind that the  is not a part of the exponent of , and is thus being multiplied to . Since  is a constant in front of , we have

.

In general, the derivative for an exponential function  is

.

In our case , , so we have

.

Hence

.

The derivative of the function is equal to 

and was found using the following rules:

, ,  , 

The derivative of the function is equal to

and was found using the following rules:

, 

Given h(x), find h'(x).

We need to derive a function composed of trigonometric terms.

Let's recall the rules

1) Derivative of sine is cosine

2) Derivative of cosine is negative sine

3) Derivative of tangent is secant squared.

Put this all together to get:

Although written correctly by convention, the superscript  that appears immediately after the trigonometric function  may obscure the problem; the function 

 is equivalent to writing .

Using the fact that

,

we apply the chain rule twice, using the power rule in the first step:

. 

(where in the last step, we have returned to the convention of writing the superscript  immediately after )

Use the chain rule:  with the outer function as  and the inner function as .

We have then:

(where we have used the chain rule again to compute the derivative of the inner function )

We can simplify this further (into the format of the answer choices) as follows:

(multiplication by a convenient form of 1)

.

Find the first derivative of the given function.

So, here we need to derive a function with trigonometric terms. Let's recall the rules

1) The derivative of secant is secant times tangent

2) The derivative of cotangent is negative cosecant squared.

3) The derivative of any constant term is 0

Put all this together to get:

And simplify to get:

The correct answer is 11.

Taking the derivative of  involves the product rule, and the chain rule.

Substituting  into both sides of the derivative we get

.