# AP Calculus AB : Derivatives

## Example Questions

### Example Question #7 : Slope Of A Curve At A Point

Find the slope of the function  at the point

Explanation:

To find the slope at a point of our function, we need to find its derivative first.

Using the product rule (and the chain rule within this product rule application), we have

.

Plugging the -value of our point into this equation, we get our desired slope of

.

### Example Question #8 : Slope Of A Curve At A Point

If , what is the slope of the curve at the point ?

Explanation:

To find the slope at a point, we first find the derivative of our function, and then substitute in the -value of our point.

, so , and plugging in our -value gives .

### Example Question #9 : Slope Of A Curve At A Point

Find the rate of change of f(x) at the point (4,12).

Explanation:

Find the rate of change of f(x) at the point (4,12)

We are asked to find a rate of change, so begin by finding the first derivative.

Now, we need the rate of change at (4,12). What matters most is the x value, simply plug it into our derivative and solve for y

### Example Question #10 : Slope Of A Curve At A Point

Find the slope of the line tangent to f(x) at the point x=0.

Explanation:

Find the slope of the line tangent to f(x) at the point x=0

To find the slope of a tangent line, we must first find the derivative of our function.

Let's recall a few rules to help us out.

1) The derivative of a monomial can be found by multiplying the coefficient by the exponent, and then decreasing the exponent by 1.

2) The derivative of e to the x is e to the x

3) The derivative of cosine is negative sine

Now, put it all together to get:

Lastly, we need to plug in 0 for x and solve our equation.

Explanation:

Explanation:

Explanation:

Explanation: