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[-/1 Points] DETAILS MY NOTES SCALCET9 3.4.059.
Find an equation of the tangent line to the curve at the given point.
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Ask by Lindsey Erickson. in the United States
Mar 20,2025

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The equation of the tangent line is .

Solución

To find the equation of the tangent line to the curve at the point , we need to follow these steps:
  1. Find the derivative of the function to determine the slope of the tangent line at the given point.
  2. Evaluate the derivative at to find the slope at that point.
  3. Use the point-slope form of the equation of a line to write the equation of the tangent line.

Step 1: Find the derivative

The function is . We will use the chain rule to find the derivative .

Step 2: Evaluate the derivative at

Let’s calculate the derivative first. The derivative of is given by:
Now, we will evaluate this derivative at .

Step 3: Use the point-slope form

The point-slope form of the equation of a line is given by:
where is the slope, and is the point on the line.
Now, let’s perform the calculations for the derivative and evaluate it at .
Calculate the value by following steps:
  • step0: Calculate:
  • step1: Calculate:
  • step2: Calculate:
  • step3: Multiply:
  • step4: Calculate:

    The derivative evaluated at is:

Step 4: Use the point-slope form

Now we have:
  • Slope
  • Point
Using the point-slope form:
This simplifies to:

Final Answer

The equation of the tangent line to the curve at the point is:

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Beyond the Answer

To find the equation of the tangent line to the curve at the point , we need to calculate the derivative of the function. The derivative, , gives us the slope of the tangent line. Evaluating at , we find .
Now, we can use the point-slope form of the line:
This simplifies to:
So, the equation of the tangent line at is:

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