To find the slope at a point of a function, take the derivative of the function.

The derivative ofis.

Therefore the derivative becomes,

since.

Now we substitute the given point to find the slope at that point.

To solve this problem, first we need to take the derivative of the function. It will be easier to rewrite the equation asfrom here we can take the derivative and simplify to get

From here we need to evaluate at the given point. In this case, only the x value is important, so we evaluate our derivative at x=2 to get.

To solve this problem, first we need to take the derivative of the function.

From here we need to evaluate at the given point. In this case, only the x value is important, so we evaluate our derivative at x=1 to get

.

To solve this problem, first we need to take the derivative of the function. To do this we need to use the quotient rule and simplified as follows:

From here we need to evaluate at the given point. In this case, only the x value is important, so we evaluate our derivative at x=2 to get

.

The derivative ofis:

Substitute the point at.

In order to find the slope of a function at a certain point, plug in that point into the first derivative of the function. Our first step here is to take the first derivative.

Since we see that f(x) is composed of two different functions, we must use the product rule. Remember that the product rule goes as follows:

Following that procedure, we setequal toandequal to.

,

which can be simplified to

.

Now plug in 1 to find the slope at x=1.

Remember that.

求解the derivative of, use implicit differentiation which means to take the derivative of each term with respect to the variable in that term.

Substitute the point into the derivative.

We must take the derivativebecause that will give us the slope. On the left side we'll get

, and on the right side we'll get.

We include theon the left side becauseis a function of, so its derivative is unknown (hence we are trying to solve for it!).

Now we can factor out aon the left side to get

and divide byin order to solve for.

Doing this gives you

.

We want to find the slope at, so we can sub inforand.

.

Here, we must use the product rule.

Assumingand, our expression becomes

.

The question asks us to evaluate this at.

As slope is defined as the derivative of a given function at a given point, we will need to take the derivative ofand substitute in the-value of the point.

Using the Power Rulefor all,. Subbing in,.