If f(x)=(3x)/(1+x^2)
find f′(4).
Use this to find the equation of the tangent line to the curve y=3x1+x2 at the point (4,0.70588). The equation of this tangent line can be written in the form y=mx+b where m is:
and where b is:
f(x) = 3x/(1+x²)
f'(x) = [(3)(1+x²) - (3x)(2x)]/(1+x²)²
= (3+3x²-6x²)/(1+x²)² = (3-3x²)/(1+x²)²
f(4) = 12/17 = 0.70588
f'(4) = -9/289 = -0.03114
So, we want a line through (4/0.70588) with slope -0.03114
(y-0.70588)/(x-4) = -0.03114
y = -0.3114x + 0.83044
Well, well, well, let's find the derivative first, shall we?
To find f'(x), we need to use the quotient rule. Remember the quotient rule? It's like mixing oil and water, they don't mix well, but we have to give it a shot.
Using the quotient rule, we have:
f'(x) = [(1+x^2) * (3) - (3x) * (2x)] / (1+x^2)^2
After simplifying, we get:
f'(x) = (3 - 6x^2) / (1+x^2)^2
Now, to find f'(4), we just substitute x = 4 into the equation:
f'(4) = (3 - 6(4^2)) / (1+4^2)^2
f'(4) = (3 - 6(16)) / (1+16)^2
f'(4) = (3 - 96) / 17^2
f'(4) = -93 / 289
Now, let's move on to the equation of the tangent line. We have the point (4, 0.70588) already, and we just found that f'(4) = -93 / 289.
The equation of the tangent line can be written in the form y = mx + b. So, m is the slope of the tangent line, which is -93 / 289, and b is the y-intercept.
Therefore, the equation of the tangent line is y = (-93 / 289)x + b. Oh, I see you're also curious about b. Well, we just need to plug in the coordinates of the given point (4,0.70588) into our equation. So,
0.70588 = (-93 / 289)(4) + b
Now it's time for some arithmetic fun to find b. Once you have it, you can enjoy the complete equation of the tangent line. Have fun crunching those numbers!
To find f'(x), we will differentiate the function f(x) with respect to x using the quotient rule.
f(x) = (3x)/(1 + x^2)
Let's differentiate f(x) step-by-step:
1. Apply the quotient rule, which states that for a function h(x) = g(x)/k(x), the derivative h'(x) is given by:
h'(x) = (g'(x) * k(x) - k'(x) * g(x)) / k(x)^2
In our case, g(x) = 3x and k(x) = 1 + x^2.
2. Find g'(x) and k'(x):
g'(x) = 3 (since the derivative of x with respect to x is 1)
k'(x) = 2x (by applying the power rule)
3. Substitute the values into the quotient rule:
f'(x) = ((3 * (1 + x^2)) - (2x * 3x)) / (1 + x^2)^2
Simplifying further:
f'(x) = (3 + 3x^2 - 6x^2) / (1 + x^2)^2
= (3 - 3x^2) / (1 + x^2)^2
Now, to find f'(4), we substitute x = 4 into the derivative:
f'(4) = (3 - 3(4)^2) / (1 + (4)^2)^2
= (3 - 3(16)) / (1 + 16)^2
= (3 - 48) / (17^2)
= -45 / 289
Therefore, f'(4) is equal to -45/289.
To find the equation of the tangent line to the curve y = 3x^1 + x^2 at the point (4, 0.70588), we can use the point-slope form of a line:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope.
Using the point (4, 0.70588) and the slope m = -45/289, we have:
y - 0.70588 = (-45/289)(x - 4)
Simplifying further:
y - 0.70588 = (-45/289)x + (45/72)
y - 0.70588 = (-45/289)x + (5/8)
Rearranging the equation to match the form y = mx + b:
y = (-45/289)x + (5/8) + 0.70588
Therefore, the equation of the tangent line is y = (-45/289)x + (13/8).
To find f'(x), the derivative of f(x), we can use the quotient rule. The derivative of f(x) = (3x) / (1 + x^2) can be found as follows:
1. Apply the quotient rule:
f'(x) = [(1 + x^2) * (3) - (3x) * (2x)] / (1 + x^2)^2
2. Simplify the expression:
f'(x) = (3 + 3x^2 - 6x^2) / (1 + x^2)^2
= (3 - 3x^2) / (1 + x^2)^2
Now, to find f'(4), we substitute x = 4 into the derived expression:
f'(4) = (3 - 3(4)^2) / (1 + (4)^2)^2
= (3 - 3(16)) / (1 + 16)^2
= (3 - 48) / (17)^2
= -45 / 289
Therefore, f'(4) = -45 / 289.
To find the equation of the tangent line to the curve y = 3x^(1 + x^2) at the point (4, 0.70588), we can use the point-slope form of a line.
1. Begin with the point-slope form: y - y1 = m(x - x1).
2. Plug in the given values: (x1, y1) = (4, 0.70588).
y - 0.70588 = m(x - 4).
3. Substitute the value of m = f'(4) into the equation:
y - 0.70588 = (-45 / 289)(x - 4).
Therefore, the equation of the tangent line can be written in the form y = mx + b, where m = -45 / 289 and b = 0.70588.