I am confused how to derivative this...
2y=x^2+siny
2 dy/dx = 2x + cosy dy/dx
2dy/dx - cosy dy/dx = 2x
dy/dx(2 - cosy) = 2x
dy/dx = 2x/(2 - cosy)
btw, the noun is "derivative"
the verb is "differentiate"
e.g.
Sissy differentiated the equation and found the derivative to be quite simple.
Well, calculating derivatives can be a bit tricky sometimes, just like trying to juggle rubber chickens while riding a unicycle. But don't worry, I'm here to help!
To find the derivative of the given equation 2y = x^2 + sin(y), we can use the chain rule, which is like finding the funny side of a clown's face.
First, let's differentiate both sides of the equation with respect to x. On the left side, the derivative of 2y with respect to x is simply 2dy/dx (since y depends on x), and on the right side, we have to apply the chain rule.
The derivative of x^2 with respect to x is 2x (just like how a clown's shoes come in pairs), and the derivative of sin(y) with respect to y is cos(y) (because clowns always have a cheerful demeanor).
However, since we are differentiating with respect to x, we need to multiply the derivative of sin(y) with respect to y (cos(y)) by dy/dx.
So, putting it all together, we get:
2(dy/dx) = 2x + cos(y) * dy/dx.
Now, let's isolate dy/dx by moving the cos(y) * dy/dx term to the left side:
2(dy/dx) - cos(y) * dy/dx = 2x.
Combining like terms, we have:
(2 - cos(y)) * dy/dx = 2x.
Finally, we can solve for dy/dx by dividing both sides by (2 - cos(y)):
dy/dx = 2x / (2 - cos(y)).
And there you go! The derivative of 2y = x^2 + sin(y) with respect to x is dy/dx = 2x / (2 - cos(y)). Just remember that math can be as funny as a clown, so don't forget to laugh along the way!
To find the derivative of the given equation 2y = x^2 + sin(y), we need to find the derivative of each term individually using the rules of differentiation. Here's how you can do it step-by-step:
Step 1: Rewrite the equation using the implicit differentiation notation by adding d/dx for each term:
2y = x^2 + sin(y)
Differentiating with respect to x, we get:
d/dx(2y) = d/dx(x^2) + d/dx(sin(y))
Step 2: Differentiate each term separately using the appropriate rules:
For the left-hand side (2y):
Using the Chain Rule, we differentiate y with respect to x, multiplied by the derivative of 2 (which is zero):
2 * dy/dx = 0 [because the derivative of a constant is zero]
Simplifying this, we get:
dy/dx = 0
For the first term on the right-hand side (x^2):
Using the Power Rule, we bring down the exponent and multiply it by the coefficient:
d/dx(x^2) = 2x
For the second term on the right-hand side (sin(y)):
Since sin(y) involves a dependent variable y, we need to use the Chain Rule. The Chain Rule states that if we have a function inside another function, we need to multiply the derivative of the outer function by the derivative of the inner function.
In this case, the outer function is sin(y), and the inner function is y. The derivative of sin(x) with respect to x is cos(x). So, we multiply by the derivative of y with respect to x (dy/dx):
d/dx(sin(y)) = cos(y) * dy/dx
Step 3: Substitute the derivatives back into the equation:
2 * dy/dx = 2x + cos(y) * dy/dx
Step 4: Solve for dy/dx (the derivative):
Rearranging the equation, we isolate dy/dx terms on one side:
2 * dy/dx - cos(y) * dy/dx = 2x
Factoring out dy/dx:
dy/dx * (2 - cos(y)) = 2x
Finally, solve for dy/dx by dividing both sides by (2 - cos(y)):
dy/dx = 2x / (2 - cos(y))
This is the derivative of the given equation.
To find the derivative of the given function, 2y = x^2 + sin(y), with respect to x, we'll need to use the chain rule and product rule.
Step 1: Rewrite the equation in terms of y:
2y = x^2 + sin(y)
Divide both sides by 2:
y = (1/2)x^2 + sin(y)/2
Step 2: Apply the chain rule to the second term:
The derivative of sin(y) with respect to x is given by: d/dx(sin(y)) = cos(y) * dy/dx
Step 3: Take the derivative of both sides of the equation with respect to x:
d/dx(y) = d/dx[(1/2)x^2 + sin(y)/2]
dy/dx = (d/dx)(1/2)x^2 + (d/dx)(sin(y)/2)
Step 4: Simplify the derivative:
dy/dx = (1/2)(d/dx)(x^2) + (1/2)(d/dx)(sin(y))
dy/dx = (1/2)(2x) + (1/2)(cos(y) * dy/dx)
Step 5: Rearrange the equation to isolate dy/dx:
dy/dx - (1/2)(cos(y) * dy/dx) = x
To combine like terms:
dy/dx * (1 - (1/2)cos(y)) = x
Finally, solve for dy/dx:
dy/dx = x / (1 - (1/2)cos(y))
Thus, the derivative of 2y = x^2 + sin(y) with respect to x is dy/dx = x / (1 - (1/2)cos(y)).