To find the equation of the tangent line to the curve at the point , we need to follow these steps:
Find the derivative of the function to determine the slope of the tangent line at the given point.
Evaluate the derivative at to find the slope at that point.
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 .