Math Problem Statement
Find the equation of the tangent line to the graph of the function f(x)=x+3/x−10 at the point (0,−(3/10)).
Enter the exact answer.
Solution
To find the equation of the tangent line to the graph of the function at the point , follow these steps:
Step 1: Find the derivative of
The slope of the tangent line at any point on the curve is given by the derivative of the function at that point. To differentiate , we can apply the quotient rule.
The quotient rule is given by: where:
Now, compute the derivatives of and :
Applying the quotient rule: Simplifying the numerator:
Step 2: Find the slope of the tangent line at
To find the slope of the tangent line at the point , substitute into the derivative:
So, the slope of the tangent line at is .
Step 3: Use the point-slope form to find the equation of the tangent line
The point-slope form of the equation of a line is: where is the slope, and is a point on the line.
We know:
Substitute these values into the point-slope form: Simplifying: Now, solve for :
Final Answer:
The equation of the tangent line is:
Would you like any further details or clarifications on this?
Here are five related questions:
- How do you apply the quotient rule for more complicated functions?
- What is the point-slope form of the equation of a line and when should it be used?
- How would you find the equation of the tangent line if the function was given in implicit form?
- Can you differentiate rational functions using the product rule instead of the quotient rule?
- How do you find the slope of a tangent line for trigonometric functions?
Tip: Always double-check the calculations when applying the quotient rule—sign errors can happen, especially when simplifying terms.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Tangent Line
Quotient Rule
Point-Slope Form
Formulas
f'(x) = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2
Point-Slope Form: y - y1 = m(x - x1)
Theorems
Quotient Rule
Point-Slope Form of a Line
Suitable Grade Level
Grades 11-12
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