Find the equation of the tangent line to the graph of y=3cot^4x at x=pi/4
3cot^4(pi/4)=3(1)=3
(pi/4,3)=(x,y)
d/dx 3cot^4x= 12cot^3x
d/dx cotx= -csc^x
d/dx x=1
(12cot^3x)(-csc^2x)(1)
m=-12cot^3xcsc^2x
equation to the tangent line would be
y-3=-12cot^3xcsc^2x(x-pi/4) ?
pretty good, but you know that
cot π/4 = 1
csc π/4 = √2, so
m = -12(1)(2) = -24
So the line is
y-3 = -24(x-π/4)
okay, I forgot to break down the equation, thank you.
To find the equation of the tangent line to the graph of y = 3cot^4x at x = pi/4, we need to find the derivative of the function at that point.
The derivative of y = 3cot^4x can be found using the chain rule. First, let's rewrite the function as y = 3cot(x)^4.
To differentiate y with respect to x, we need to differentiate each term. The derivative of 3 is 0 since it is a constant.
Next, we differentiate cot(x)^4 using the chain rule. The derivative of cot(x) is -csc^2(x), and since we have (cot(x))^4, we can apply the power rule. The derivative of (cot(x))^4 is 4(cot(x))^3 * (-csc^2(x)).
So, the derivative of y = 3cot(x)^4 is dy/dx = 4(3cot^3(x))( -csc^2(x)) = -12cot^3(x)csc^2(x).
Now, let's find the value of the derivative at x = pi/4. Plugging in x = pi/4 into the derivative, we get:
dy/dx = -12cot^3(pi/4)csc^2(pi/4).
cot(pi/4) = 1 and csc(pi/4) = √2.
Substituting these values, we have:
dy/dx = -12(1)^3 (√2)^2 = -12(1)(2) = -24.
So, the slope of the tangent line at x = pi/4 is -24.
The equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-intercept. We know the slope is -24, and to find the y-intercept, we need to find the value of y at x = pi/4.
Plugging x = pi/4 into the original function, we have:
y = 3cot^4(pi/4) = 3(1) = 3.
So, the point on the graph at x = pi/4 is (pi/4, 3).
Therefore, the equation of the tangent line at x = pi/4 with a slope of -24 and passing through the point (pi/4, 3) is:
y - 3 = -24(x - pi/4).
Simplifying, we get:
y - 3 = -24x + 6pi.
Finally, rearranging, we have:
y = -24x + (6pi + 3).
Thus, the equation of the tangent line to the graph of y = 3cot^4x at x = pi/4 is y = -24x + (6pi + 3).