To find the derivative of this function, we need to use the Fundemental Theorem of Calculus Part 1 (As opposed to the 2nd part, which is what's usually used to evaluate definite integrals)

. Start

. Take derivatives of both sides.

. "Cancel" the integral and the derivative. (Make sure that the upper bound on the integral is a function of , and that the lower bound is a constant before you cancel, otherwise you may need to use some manipulation of the bounds to make it so.)

The rate of change of a function at a point is the value of the derivative at that point. First, take the derivative of f(x) using the power rule for each term.

Remember that the power rule is 

, and that the derivative of a constant is zero.

Next, notice that the x-value of the point (1,6) is 1, so substitute 1 for x in the derivative.

Therefore, the rate of change of f(x) at the point (1,6) is 14. 

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)

.