Find the derivative by solving the given implicit equation for y explicitly in terms of x.

Question

Find the derivative  by solving the given implicit equation for y explicitly in terms of x.

Quizplus · Accepted Answer

The given equation is not in a standard form, so we first need to use the chain rule to find the derivative:

$\frac{d}{dx}\left[11e^{xy}\right] = \frac{d}{dx}\left[11e^{xy}\right] = 11y e^{xy} \frac{dy}{dx} + 11xe^{xy}$

Setting this equal to 0 (since we want to solve for the derivative when y = f(x)), we get:

$11y e^{xy} \frac{dy}{dx} + 11xe^{xy} = 0$

Solving for $\frac{dy}{dx}$, we get:

$\frac{dy}{dx} = -\frac{x}{y}$

Now we need to find an expression for y in terms of x. To do this, we can rearrange the original equation:

$11e^{xy} = y^3 - x^3$

$e^{xy} = \frac{y^3 - x^3}{11}$

$xy = \ln{\left(\frac{y^3 - x^3}{11}\right)}$

$y = \frac{1}{x}\ln{\left(\frac{y^3 - x^3}{11}\right)}$

So the derivative when y = f(x) is:

$\frac{dy}{dx} = -\frac{x}{y} = -\frac{x}{\frac{1}{x}\ln{\left(\frac{y^3 - x^3}{11}\right)}} = -\frac{x^2}{\ln{\left(\frac{y^3 - x^3}{11}\right)}}$

Plugging in y = f(x), we get:

$\frac{dy}{dx} = -\frac{x^2}{\ln{\left(\frac{\left(\frac{1}{x}\ln{\left(\frac{f(x)^3 - x^3}{11}\right)}}\right)^3 - x^3}{11}\right)}}$

Simplifying this expression gives:

$\frac{dy}{dx} = -\frac{x^2}{\ln{\left(\frac{f(x)^3}{x^3} - 1\right)} - \ln{11}}$

So the answer is choice (B).

Find the Derivative by Solving the Given Implicit Equation