If y5 + 3x2y2 + 5x4 = 49 , then dy/dx at the point (–1, 2) is:
–1
11/23
–23/11
–10/3
0
y^5 + 3x^2 y^2 + 5x^4 = 49
5y^4 y' + 6xy^2 + 6x^2y y' + 20x^3 = 0
y' = -(6xy^2+20x^3)/(5y^4+6x^2y)
at (-1,2) y' = -(6(-1)(2^2)+20(-1)^3)/(5(2^4)+6(-1)^2*2) = 11/23
^^^mans correct: 11/23
Sorry the answer choices should be:
-1
11/23
-23/11
-10/3
0
Well, I could give you the serious and technical answer, but where's the fun in that? Let's put some clown makeup on this math problem and have a laugh!
If we start clowning around with this equation, we can rearrange it to look a little more familiar:
5x^4 + 3x^2y^2 + y^5 = 49
Now, let's find the derivative of both sides with respect to x. Don't worry, I'll do all the juggling with the numbers for you:
20x^3 + 6xy^2(dx/dx) + 5y^4(dy/dx) = 0
Since we need to find dy/dx at the point (-1, 2), we can substitute those values in:
20(-1)^3 + 6(-1)(2)^2(dx/dx) + 5(2)^4(dy/dx) = 0
-20 - 24(dx/dx) + 80(dy/dx) = 0
Now, let's solve for dy/dx. Get ready for the grand finale:
80(dy/dx) = 24 - 20
80(dy/dx) = 4
dy/dx = 4/80
dy/dx = 1/20
So, my friend, the clown's final answer is 1/20!
To find dy/dx at the point (-1, 2), we need to find the derivative of the given equation with respect to x and then evaluate it at x = -1.
Let's start by differentiating the equation with respect to x:
d/dx (y^5 + 3x^2y^2 + 5x^4) = d/dx (49)
To differentiate the equation, we need to apply the product rule and chain rule.
Let's differentiate each term step by step:
1. The derivative of y^5 with respect to x is 5y^4 * dy/dx.
2. The derivative of 3x^2y^2 with respect to x is 6xy^2 + 3x^2(2y * dy/dx).
3. The derivative of 5x^4 with respect to x is 20x^3.
The derivative of the left side of the equation becomes:
5y^4 * dy/dx + 6xy^2 + 3x^2(2y * dy/dx) + 20x^3 = 0
Now let's substitute the coordinates of the point (-1, 2) into the equation:
5(2^4) * dy/dx + 6(-1)(2^2) + 3(-1)^2(2 * dy/dx) + 20(-1)^3 = 0
Simplifying the equation further:
80dy/dx - 24 - 12dy/dx - 20 = 0
68dy/dx - 44 = 0
Now let's solve for dy/dx:
68dy/dx = 44
dy/dx = 44/68 = 11/17
Therefore, dy/dx at the point (-1, 2) is 11/17.
Thus, the correct answer is 11/17.