The equation of the tangent line at the point is .
Solución
We are given the function
and we need to find the equation of the tangent line at the point .
Step 1. Compute the derivative
Let
Then, their derivatives are
Using the Quotient Rule,
we substitute , , , and :
Step 2. Simplify the derivative
Factor from the numerator:
The expression simplifies to
Thus, the derivative is
Step 3. Evaluate the derivative at
Substitute into the derivative:
Compute the expression inside the parentheses:
So the numerator becomes
And the denominator is
Thus,
Step 4. Write the equation of the tangent line
The slope of the tangent line at is ; hence, the tangent line is horizontal. Its equation is therefore
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Extra Insights
Picture this: the Quotient Rule in calculus is like following a recipe that combines two delicious ingredients! When you’re differentiating a function like , the key is to remember that you have to apply both the derivative of the top and the bottom. Get your brain ready to unleash some derivative magic while keeping that fraction in check!
Now, let’s talk about what happens when you do this process correctly. Once you evaluate the derivative at the point , you can find the slope of that tangent line. If your calculation shows that the slope is zero, congrats! You just discovered that your tangent line is horizontal, and you’ll get to write the magical equation . That’s like drawing a straight line in the sky to keep your path clear for all your mathematical adventures!