Find d2y/dx2 in terms of x and y.
y5 = x6
Well, I could definitely make a "derivative" joke here, but I wouldn't want to "differentiate" your interest in math. So let's get straight to it!
To find d²y/dx² (the second derivative of y with respect to x) in terms of x and y, we'll need to differentiate both sides twice.
Starting with the equation y^5 = x^6, let's do some math magic:
First, differentiate both sides with respect to x, using the chain rule:
(5y^4)(dy/dx) = (6x^5)
To make this a bit more manageable, let's rewrite it as:
dy/dx = (6x^5) / (5y^4)
Now, d²y/dx² is just the derivative of dy/dx with respect to x. Applying the quotient rule, we get:
d²y/dx² = [(5y^4)(30x^4) - (6x^5)(20y^3)] / (5y^4)^2
That simplifies to:
d²y/dx² = (150x^4y^4 - 120x^5y^3) / (25y^8)
And there you have it! The second derivative, d²y/dx², in terms of x and y. Hope that equation didn't "differentiate" your spirits, my friend!
To find the second derivative of y with respect to x, we'll need to first find the first derivative and then take the derivative of it.
Given: y^5 = x^6
Step 1: Take the derivative of both sides of the equation with respect to x.
d/dx(y^5) = d/dx(x^6)
Step 2: Use the power rule to differentiate y^5 and x^6.
5y^4 * (dy/dx) = 6x^5
Step 3: Solve for dy/dx (the first derivative).
(dy/dx) = 6x^5 / (5y^4)
Step 4: Take the derivative of the equation found in step 3 with respect to x to find the second derivative (d^2y/dx^2).
d/dx[(dy/dx)] = d/dx[6x^5 / (5y^4)]
Step 5: Apply the quotient rule to differentiate 6x^5 / (5y^4).
(d^2y/dx^2) = [(5y^4 * d/dx(6x^5)) - (6x^5 * d/dx(5y^4))] / (5y^4)^2
Step 6: Simplify the equation further.
(d^2y/dx^2) = [30x^4 * y^4 - 120x^5 * y^3] / (25y^8)
So, the second derivative (d^2y/dx^2) in terms of x and y is (30x^4 * y^4 - 120x^5 * y^3) / (25y^8).
To find the second derivative of y with respect to x (d^2y/dx^2), we first need to differentiate the given equation, which is y^5 = x^6, with respect to x.
Differentiating both sides of the equation implicitly, we get:
5y^4 * dy/dx = 6x^5
Now, we need to solve for dy/dx by isolating it:
dy/dx = (6x^5) / (5y^4)
Next, we differentiate this expression, again with respect to x, to find d^2y/dx^2:
To do this, we use the quotient rule, which states:
If f(x) = u(x) / v(x), where u(x) and v(x) are functions of x, then:
f'(x) = (u'(x) * v(x) - u(x) * v'(x)) / v(x)^2
Applying the quotient rule to dy/dx = (6x^5) / (5y^4), we have:
d^2y/dx^2 = [(d/dx)(6x^5) * (5y^4) - (6x^5) * (d/dx)(5y^4)] / (5y^4)^2
Now, let's differentiate each term separately:
(d/dx)(6x^5) = 30x^4
(d/dx)(5y^4) = 20y^3 * dy/dx (using the chain rule)
Substituting these values back into the expression for d^2y/dx^2, we have:
d^2y/dx^2 = [30x^4 * (5y^4) - (6x^5) * (20y^3 * dy/dx)] / (5y^4)^2
Finally, we can substitute the value of dy/dx we found earlier:
d^2y/dx^2 = [30x^4 * (5y^4) - (6x^5) * (20y^3 * (6x^5 / 5y^4))] / (5y^4)^2
Simplifying this expression would yield the second derivative d^2y/dx^2 in terms of x and y.
y = x^(6/5)
dy/dx = (6/5)*y^1/5
d2y/dx^2 = (6/25)y^(-4/5)