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 Given:

Implement the sin^-1 derivative:

Understand that we find our chained term by deriving the compound element(in this case, x^3)

Plug our new-found element in for the chain, and multiply accordingly:

After multiplying, we arrive at the correct answer:

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 Given:

Apply the derivative of arc-tangent (tan^-1)

Understand that to find the (chain term), we must derive the compound function element, in this case it is (1-r^2)

Now, we plug -2r in for the chain term

Multiplying it across, we arrive at the final, correct answer:

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The derivative of the function is equal to

and was found using the following rules:

, , , 

Note that because  is a function of x, when we use the product rule, we must include its derivative.

We know how to take the derivative of , but not , so let's use the chain rule.

According to the chain rule, we should take the derivative of the outside function and multiply it by the derivative of the inside function. This gives us:

Remember that .

Our final answer:

There are two ways to do this problem.

Method 1: Use the power rule to turn into , the derivative of which is .

Method 2: Use the chain rule.

Either way, the answer is .

Using the chain rule :

First, use the chain rule :

Now we have to use the product rule

Our final answer:

This problem requires us to use the chain rule twice.

First, take the derivative of .

Now substitute for and multiply by the derivative of .

is our final answer.

The slope of the tangent line is found by taking the derivative. In this case, because we don't have the function defined explicitly, we'll have to use implicit differentiation to get

Where the first two terms come from product rule. Factoring out a dy/dx and moving the other terms to the right, we get

And solving for dy/dx, we get

.

As the problem doesn't give us a y value, we'll need to go back to the original equation to find it. Plugging in x = 1, we have 

And simplifying yields

Plugging the point (1, 3) into our expression for dy/dx, we have

Alternatively, this expression could have been solved for explicitly by factoring a y term out. Note,

And thus, . Using quotient rule, we know the derivative of this

is .

Evaluated at x=1, this becomes

Finding for an equation of this complexity requires implicit differentiation.

We need to take the derivative of both sides with respect to x.

For , we will apply the product rule and chain rule.

The product rule is as follows: , where f and g are the two factors of the expression. In this case, and will be and respectively. When we find , we will use the chain rule. This is because the inside of the contains variables other than x. Recall that the derivative of  with respect to  is . 

This all means that

Observe that  which is just 1.

Applying all this we find that the derivative of is

.

Now we will find the derivative of the next term, . This will also require product rule and chain rule. For this product rule,  and  will be  and  respectively.

will be the derivative of with respect to , which is .

will require the chain rule. The derivative of is , and the derivative of  with respect to is .

This means that .

Assembling the power rule gives the derivative of the second term as

For the right hand side of the equation, the derivative of is still .

Putting all three derivatives together into one equation yields the following:

The next step in implicit differentiation is solving for . To do this we move all terms that have no factor to one side, and all terms with a factor to the other side.

In this case, we already have all terms on the left, so we will move the non- terms to the right side.

Now we can pull the common factor of out of the left side. We will then divide by what is remaining.

This is the final answer. It cannot be simplified.