*find the slope of y= 4/x +6
at the point (8,2)
*evaluate the derivative at the point (pi,4)
y=[tan(5x^8)]^3
please help
just tel me how to setup
I will assume your equation is as you typed it, and not
y = 4/(x+6)
y = 4/x + 6
= 4x^-1 + 6
dy/dx = -4x^-2
= -4/x^2
at (8,2) , dy/dx = -4/64 = -1/16
equation:
y-2 = (-1/16)(x - 8)
16y - 32 = -x + 8
x + 16y = 40
y = [tan(5x^8)]^3
dy/dx = 3[tan(5x^8)]^2 (sec(5x^8) (40x^7)
at (π/4)
use your calculator to find the messy answer for dy/dx
-4/64
why 64 not 8
notice dy/dx = -4/(x^2)
= -4/(8^2) = -4/64 or -1/16
just plug pi/4
right
thnks Reiny.
To find the slope of a function at a specific point, you need to calculate the derivative of the function and then substitute the coordinates of the point into the derivative expression.
1. Find the derivative of y = 4/x + 6:
To do this, you can use the power rule and the constant rule for differentiation.
Let's rewrite the equation as y = 4x^(-1) + 6:
dy/dx = d/dx(4x^(-1)) + d/dx(6)
= -4x^(-2) + 0 (since the derivative of a constant is 0)
= -4/x^2
2. Evaluate the derivative at the point (8,2):
Substitute x = 8 into the derivative expression:
dy/dx = -4/8^2
= -4/64
= -1/16
Therefore, the slope of the function y = 4/x + 6 at the point (8,2) is -1/16.
To evaluate the derivative of y = [tan(5x^8)]^3 at the point (pi, 4):
1. Find the derivative of y = [tan(5x^8)]^3:
To do this, we can use the chain rule and power rule for differentiation.
Let's rewrite the equation as y = (tan(5x^8))^3:
dy/dx = 3(tan(5x^8))^2 * d/dx(tan(5x^8))
Now, we need to differentiate the expression inside the tangent function:
d/dx(tan(5x^8)) = sec^2(5x^8) * d/dx(5x^8)
= 5sec^2(5x^8) * d/dx(x^8)
= 5sec^2(5x^8) * (8x^7)
Thus, the derivative of y = [tan(5x^8)]^3 is:
dy/dx = 3(tan(5x^8))^2 * 5sec^2(5x^8) * 8x^7
= 120x^7(tan(5x^8))^2 sec^2(5x^8)
2. Evaluate the derivative at the point (pi, 4):
Substitute x = pi into the derivative expression:
dy/dx = 120(pi)^7(tan(5(pi)^8))^2 sec^2(5(pi)^8)
After calculating the expression with pi as the value of x, you will get the value of the derivative at the point (pi, 4).