Qube said:Homework Statement
http://i.minus.com/jDtSwpCGlrMhP.jpg
Homework Equations
Solving for dx/dy (change in x with respect to y) I get a solution that isn't one of the answer choices.
The Attempt at a Solution
3y^2 = 8x' + x'/x
Coordinate:
x = 1 (given)
y = 2 (solved for in original equation)
3(4) = 9x'
12/9 = dx/dy
4/3 = dx/dy (change of x with respect to y).
brmath said:No matter what derivatives you take, please make sure you apply the chain rule properly. I don't think you did that in your attempt.
Mark44 said:The tipoff that they're looking for time rates of change (dx/dt and dy/dt) is the units in the answers - units/sec.
Qube said:Thank you. That makes more sense. I found dx/dt and plugged in dy/dt which is -1/2.
3y^2(dy/dt) = 8(dx/dt) + dx/dt*(1/x)
y=2, as always.
3(4)(-1/2) = 9(dx/dt)
-6 = 9(dx/dt)
-2/3 = dx/dt
It appears the answer is D.
Differentiation is a mathematical operation that measures the rate at which one quantity changes with respect to another quantity. It is important in many fields of science, including physics, engineering, and economics, as it allows us to analyze and understand the behavior of changing systems.
In Calculus III, differentiation involves finding the partial derivatives of functions with multiple variables. This means that instead of finding the rate of change along a single axis, we are finding the rate of change along multiple axes simultaneously. This requires a deeper understanding of multivariable calculus concepts such as gradients and directional derivatives.
Differentiation has many practical applications in fields such as physics, engineering, and economics. For example, in physics, differentiation is used to calculate the velocity and acceleration of moving objects. In economics, it is used to analyze supply and demand curves. In engineering, it is used to optimize designs and predict the behavior of complex systems.
Yes, differentiation can be used to find the maximum and minimum values of a function. In fact, one of the main applications of differentiation is optimization, which involves finding the maximum or minimum value of a function. This is done by setting the derivative of the function equal to zero and solving for the critical points.
There are several methods for differentiating functions, including the power rule, product rule, quotient rule, and chain rule. The power rule is used for functions with a single variable raised to a power. The product rule is used for functions that are the product of two other functions. The quotient rule is used for functions that are the quotient of two other functions. And the chain rule is used for composite functions, where one function is nested inside another.