Find the equation of the tangent line to the graph of the function
f(x)=x-3/3x-5 at x=1
f'(x) = 4/(3x-5)^2
f'(1) = 4/4 = 1
f(1) = -2/-2 = 1
now you have a point (1,1) and a slope (1)
(y-1) = 1(x-1)
To find the equation of the tangent line to the graph of the function f(x) at x=1, we need to find the slope of the tangent line and the point where it intersects the graph.
Step 1: Find the derivative of the function f(x).
Let's differentiate f(x)=x-3/3x-5 using the quotient rule.
f'(x) = (1 * (3x-5) - (x-3) * 3) / (3x-5)^2
= (3x - 5 - 3x + 9) / (3x-5)^2
= 4 / (3x-5)^2
Step 2: Evaluate the derivative at x=1 to find the slope of the tangent line.
Plugging x=1 into the derivative, we have:
f'(1) = 4 / (3(1)-5)^2
= 4 / (-2)^2
= 4 / 4
= 1
So, the slope of the tangent line is 1.
Step 3: Find the point where the tangent line intersects the graph.
To find the y-coordinate of this point, we substitute x=1 into the original function:
f(1) = 1 - 3 / 3(1) - 5
= 1 - 3 / 3 - 5
= 1 - 3 / -2
= 1 + 3 / 2
= 1 + 1.5
= 2.5
So, the point where the tangent line intersects the graph is (1, 2.5).
Step 4: Use the point-slope form to write the equation of the tangent line.
Using the point-slope form, the equation of the tangent line is:
y - y1 = m(x - x1)
where (x1, y1) is (1, 2.5) and m is the slope, which is 1. Plugging these values in, we get:
y - 2.5 = 1(x - 1)
y - 2.5 = x - 1
y = x + 0.5
Therefore, the equation of the tangent line to the graph of f(x) at x=1 is y = x + 0.5.
To find the equation of the tangent line to the graph of a function at a specific point, you need to follow these steps:
1. Find the derivative of the function.
2. Evaluate the derivative at the given point to find the slope of the tangent line.
3. Substitute the point coordinates and slope into the equation of a straight line to find the equation of the tangent line.
Let's go through these steps one by one for the given function, f(x) = (x - 3) / (3x - 5), and the point x = 1:
1. Find the derivative of the function:
The derivative is the rate at which the function is changing with respect to x. To find the derivative of f(x), you can use the quotient rule. The quotient rule states that if you have a function f(x) = g(x) / h(x), then the derivative of f(x) is (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2.
In our case, g(x) = x - 3 and h(x) = 3x - 5. Applying the quotient rule, we have:
f'(x) = [(3x - 5)(1) - (x - 3)(3)] / (3x - 5)^2
Expanding and simplifying, we get:
f'(x) = (3x - 5 - 3x + 9) / (3x - 5)^2
= 4 / (3x - 5)^2
2. Evaluate the derivative at x = 1:
Substitute x = 1 into the derivative expression:
f'(1) = 4 / (3(1) - 5)^2
= 4 / (-2)^2
= 4 / 4
= 1
So, the slope of the tangent line at x = 1 is 1.
3. Substitute the point coordinates and slope into the equation of a straight line:
The equation of a straight line is given by y = mx + b, where m is the slope and b is the y-intercept. We already have the slope, which is 1. Now, we need the y-coordinate of the point to find the y-intercept.
Substitute x = 1 into the original function:
f(1) = (1 - 3) / (3(1) - 5)
= (-2) / (-2)
= 1
So, the point on the function f(x) at x = 1 is (1, 1).
Using the point (1, 1) and the slope m = 1, we can write the equation of the tangent line as:
y - y1 = m(x - x1)
y - 1 = 1(x - 1)
y - 1 = x - 1
y = x
Therefore, the equation of the tangent line to the graph of f(x) = (x - 3) / (3x - 5) at x = 1 is y = x.