AP Calculus abdominal Help » Derivatives » Derivative in ~ a point » steep of a curve in ~ a point

Find the slope of the curve

at the suggest with x-coordinate
.

You are watching: Slope of a curve at a point

Explanation:

To discover the slope in ~ a point, we take the derivative of

, and also simplify.

.

.

Explanation:

First, use the chain ascendancy to uncover f"(t).

You need to get

.

Next, plugin t=0 to gain f"(0)=

.

Explanation:

Find the derivative of the curve using the strength rule.

In mathematical terms, the power preeminence states,

Therefore the derivative is,

Next, plug in the x-value to discover the steep of the curve at

, which provides you a last answer of

Explanation:

Expand the binomial and also combine prefer terms to get,

Next, take the derivative of the polynomial utilizing the strength rule, which is in math terms,

Therefore,

Lastly, plug in

for
, to get the slope at the point.

Explanation:

Derivatives space slope finders. Thus to uncover the slope at the provided point, we need to discover the derivative of the duty using strength rule. Power ascendancy says that us take the exponent that the “x” value and bring it to the front. Then us subtract one native the exponent.

So, us get

From there we plugin our x value.

Explanation:

To uncover the tangent heat at the given point, we need to very first take the derivative of the given function.

To find the derivative we must use product rule. Product preeminence states that us take the derivative of the very first function and also multiply the by the derivative that the second role and then add that with the derivative that the second duty multiplied by the given an initial function. To discover the derivative of each seperate duty we must use power rule.

Power rule says that us take the exponent of the “x” value and also bring it to the front. Then us subtract one native the exponent

Use power rule and we gain :

From here, to find the slope at the given allude we plug in "2" because that x.

This comes the end to equal

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### Example inquiry #1 : steep Of A Curve at A suggest

Find the steep of the function

at the point

None that the various other answers

Explanation:

To find the slope in ~ a allude of our function, we require to find its derivative first.

Using the product ascendancy (and the chain rule within this product dominance application), us have

.

Plugging the

-value the our suggest into this equation, we gain our wanted slope of

.

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### Example concern #8 : slope Of A Curve in ~ A suggest

If

, what is the steep of the curve in ~ the point
?

Explanation:

To find the slope in ~ a point, we first find the derivative of ours function, and also then instead of in the

-value of our point.

, so
, and also plugging in our
-value gives
.

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### Example question #9 : slope Of A Curve at A suggest

Find the price of readjust of f(x) in ~ the suggest (4,12).

Explanation:

Find the price of adjust of f(x) in ~ the suggest (4,12)

We space asked to find a price of change, so begin by recognize the very first derivative.

Now, we require the rate of adjust at (4,12). What matters many is the x value, merely plug it right into our derivative and solve because that y

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### Example inquiry #10 : steep Of A Curve at A suggest

Find the steep of the line tangent come f(x) at the point x=0.

Explanation:

Find the steep of the heat tangent come f(x) in ~ the point x=0

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

Let"s recall a couple of rules to help us out.

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

See more: What Is A Baby Goose Called ? Facts You Should Know What Is A Baby Goose Called

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

3) The derivative the cosine is an unfavorable sine

Now, placed it all with each other to get:

Lastly, we should plug in 0 because that x and solve our equation.

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