Derivatives of equations

Question

Assume that $x$ and $y$ are related by the equation $y\ln x=e^{1−x}+y^3$. Compute $dy/dx$ evaluated at $x=1$.

I do not understand how to compute the derivative of an equation. Please explain.

Answer 1

Here are the first few steps you need to do. As others have noted, it is called implicit differentiation. It is very similar to "normal" differentiation, but the difference is that you consider $y$ to be a function of $x$. When you want to differentiate it with respect to $x$, you get that $\frac{d}{dx}(y) = \frac{dy}{dx}$ (very simply) . Also, you need to remember to apply to $y$ all of the rules you normally apply to functions of $x$. For example, you need to apply the product rule to $\frac{d}{dx}(xy)$, among other things.

So you have that

$$y\ln x = e^{1-x} + y^3$$ thus $$\frac{d}{dx} (y\ln x) = \frac{d}{dx} ( e^{1-x} + y^3 )$$ so $$\frac{dy}{dx}\ln x + \frac{y}{x} = -e^{1-x} + 3y^2 \frac{dy}{dx}$$

Now you should be able to solve for $\frac{dy}{dx}$, which will be dependent of both $x$ and $y$.

Answer 2

As the others have mentioned, your problem requires implicit differentiation. Firstly, anytime you differentiate, you are secret implicitly deriving but you just don't know it! Your problem is the following:

$$y\ln x=e^{1−x}+y^3$$

and we are required to find $\frac{dy}{dx}$ which means find the derivative with respect to $x$. This means, $y$ is a function $x$ or in other words, $y(x)$. So, let's get to differentiating.

\begin{align} y\ln x &= e^{1-x} + y^3 \\ \frac{d}{dx} y\ln x &= \frac{d}{dx} e^{1-x} + \frac{d}{dx} y^3 \\ \end{align}

Now, because $y$ is actually a function it's derivative should be $\frac{dy}{dx}$. Keeping that in mind, let's continue. On the left hand side, we'll use the product rule and on the right hand side, we'll differentiate seprately using chain rule and product rule. Finally, we'll rearrange for $\frac{dy}{dx}$.

\begin{align} \frac{d}{dx} y\ln x &= \frac{d}{dx} e^{1-x} + \frac{d}{dx} y^3 \\ \frac{dy}{dx}\ln x + y\left(\frac{1}{x}\right) &= e^{1-x} \cdot (-1) + \frac{dy}{dx}\cdot3y^2 \\ \frac{dy}{dx}\ln x - \frac{dy}{dx}3y^2 &= -\left(e^{1-x} + \frac{y}{x}\right)\\ \end{align}

I'm going to leave the last few steps for you to do but it's almost done! All you have to do is isolate $\frac{dy}{dx}$ and you're done!

If you have any questions then please don't hesitate to ask!

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Answer 3

Use the chain rule/implicit differentiation. For example, if $f(x)s(y)+g(x)+h(y)=j(y)$ is an equation ($y=q(x)$ a function of $x$), then differentiate both sides to obtain $$f^\prime(x)s(y)+f(x)s^\prime(y)y^\prime+g^\prime(x)+h^\prime(y)y^\prime=j^\prime(y)y^\prime\\ \implies f^\prime(x)s(y)+g^\prime(x)=y^\prime(j^\prime(y)-h^\prime(y)-f(x)f^\prime(y))\\ \implies y^\prime=\dfrac{f(x)s^\prime(y)+g^\prime(x)}{j^\prime(y)-h^\prime(y)-f(x)s^\prime(y)}$$ In your case, $f(x)=\ln x,s(y)=y, h(y)=-y^3, g(x)=-e^{1-x},$ and $j(y)=0$.

Answer 4

You need to do it implicitly. You noted that you don't know how to differentiate an equation, but what you want to do is find the quantity dy/dx. To do this, note that, if you differentiate both expressions, they will remain equal

Answer 5

Hint: You're not computing the derivative of an equation. You're computing the derivative of $y$ with respect to $x$. Since you aren't given $y$ in terms of $x$, your answer will most likely be in terms of both $y$ and $x$, not just $x$.

Answer 6

Here is the complete answer. First differentiate both sides of the equation $$ D(y \ln(x)) = D(e^{1-x}+y^3)\\ \to D(\ln(x))\times y+ D(y)\times \ln(x) = -e^{1-x} + D(y^3) $$ Using implicit differentiation we have $$ \frac{1}{x}y+y^{\prime} \ln(x) = -e^{1-x}+3y^2 y^{\prime} $$ Now we have $$ y^{\prime} \ln(x) - 3y^2 y^{\prime} +\frac{1}{x}y + e^{1-x} = 0 \\ \to y^{\prime}(\ln(x)-3y^2)=-\frac{1}{x}y-e^{1-x} $$ Thus $$ y^{\prime}=\frac{-y/x-e^{1-x}}{\ln(x)-3y^2} $$