1. find the equation of the tangent line at x=1 to y=f(x) where f(x)=(3x^2)/(5x^2+7x)
Can someone help walk me through the steps of this problem. Do I start with finding the derivative of the function using the quotient rule?
sorry I didn't mean to repost this. When I refreshed by computer did it automatically for whatever reason. I posted the same question below but if someone can help and wants to answer here that's fine too.
Unless there's a typo,
y = (3x)/(5x+7), so
y(1) = 1/4
y' = 21/(5x+7)^2
y'(1) = 112/363
Now you have a point and a slope, so the equation of the line is dead simple.
Yes, you are correct. To find the equation of the tangent line at x=1 to the function y=f(x), you need to start by finding the derivative of the function using the quotient rule.
Here's how you can solve this problem step-by-step:
1. Start with the given function: f(x) = (3x^2) / (5x^2 + 7x).
2. Find the derivative of this function using the quotient rule.
- Apply the quotient rule: The derivative of f(x) = (g(x) / h(x)) is given by (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2.
- Let g(x) = 3x^2 and h(x) = 5x^2 + 7x.
- Find g'(x) and h'(x) by taking the derivative of g(x) and h(x), respectively.
- g'(x) = 6x.
- h'(x) = 10x + 7.
- Now, apply the quotient rule formula: f'(x) = [(5x^2 + 7x) * (6x) - (3x^2) * (10x + 7)] / (5x^2 + 7x)^2.
- Simplify the expression for f'(x) as much as possible.
Once you have found the derivative f'(x), you can proceed to find the equation of the tangent line at x=1 using the point-slope form of a line.
3. Plug in x=1 into the derivative f'(x) to find the slope of the tangent line.
- Evaluate f'(x) by substituting x=1 into the expression for f'(x) you obtained in step 2.
4. Use the point-slope form of a line to find the equation of the tangent line.
- The point-slope form of a line is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
- You know the point (x1, y1) is (1, f(1)), and you found the slope (m) in step 3.
- Substitute the values of (x1, y1) and m into the point-slope form to obtain the equation of the tangent line.
By following these steps, you can find the equation of the tangent line at x=1 to the function y=f(x).