Find the derivative of the function, xy-x-12y-3=0.
Is the answer -14,I got this from 1-1-12=0, which solves to -14.
Don't you have to do implicit differentiation?
Do you want dy/dx or dx/dy?
The answer of -14 makes no sense at all. Do you understand what a derivative is? It is the rate of change of a function. In this case, they want a formula for it.
To find the derivative of the function, we will use implicit differentiation because the function is not given explicitly as \(y = f(x)\). Here's how you can proceed:
Step 1: Differentiate both sides of the equation with respect to \(x\).
\( \frac{d}{dx} (xy - x - 12y - 3) = \frac{d}{dx} 0 \)
Step 2: Apply the rules of differentiation. For terms involving \(y\), we need to use the chain rule.
The derivative of \(xy\) with respect to \(x\) is \(\frac{d(xy)}{dx} = y + x \frac{dy}{dx}\).
The derivative of \(-12y\) with respect to \(x\) is \(\frac{d(-12y)}{dx} = -12 \frac{dy}{dx}\).
The derivative of \(0\) with respect to \(x\) is \(0\).
Step 3: Simplify the equation.
\( y + x \frac{dy}{dx} - 1 - 12 \frac{dy}{dx} = 0 \)
Step 4: Combine like terms.
\( x \frac{dy}{dx} - 11 \frac{dy}{dx} + y - 1 = 0 \)
Step 5: Combine the differential terms.
\( (x - 11) \frac{dy}{dx} + y - 1 = 0 \)
This is the derivative of the function. The answer is not -14 but rather \((x - 11) \frac{dy}{dx} + y - 1 = 0\).
It seems there might have been an error in your calculations. Please double-check your work for evaluating the equation.