Hi there!
We can calculate dy/dx using implicit differentiation:
xy + y² = 6
Differentiate both sides. Remember to use the Product Rule for the "xy" term:
(1)y + x(dy/dx) + 2y(dy/dx) = 0
Move y to the opposite side:
x(dy/dx) + 2y(dy/dx) = -y
Factor out dy/dx:
dy/dx(x + 2y) = -y
Divide both sides by x + 2y:
dy/dx = -y/x + 2y
We need both x and y to find dy/dx, so plug in the given value of x into the original equation:
-1(y) + y² = 6
-y + y² = 6
y² - y - 6 = 0
(y - 3)(y + 2) = 0
Thus, y = -2 and 3.
We can calculate dy/dx at each point:
At y = -2: dy/dx = -(-2) / -1+ 2(-2) = -2/5.
At y = 3: dy/dx = -(3) / -1 + 2(3) = -3/5.
Answer:
3=-3
A radical is a mathematical symbol used to represent the root of a number. Here’s a quick example: the phrase “the square root of 81” is represented by the radical expression . (In the case of square roots, this expression is commonly shortened to —notice the absence of the small “2.”) When we find we are finding the non-negative number r such that , which is 9.
While square roots are probably the most common radical, we can also find the third root, the fifth root, the 10th root, or really any other nth root of a number. The nth root of a number can be represented by the radical expression.
Radicals and exponents are inverse operations. For example, we know that 92 = 81 and = 9. This property can be generalized to all radicals and exponents as well: for any number, x, raised to an exponent n to produce the number y, the nth root of y is x.
The Third Angle Theorem states if two angles are congruent to two angles in another triangle, the third angles are congruent too. Because a triangle has <span><span><span>180 degrees </span></span></span>the third angle in any triangle is <span><span><span>180 degrees </span></span></span>minus the other two angle measures
