Implicit differentiation featuring trig functions

Question

How would I solve the following question that is troubling me.

The question is

Find an equation for the tagnet line to the graph of $x+\sin(y-2x)=1$ at point $(1,2)$

I did the following using the chain

$1+\cos(y-2x)(\frac{dy}{dx})(-2)$

then I simplified to $\frac{dy}{dx}=\frac{2}{cos(y-2x)}$ so I pluged in $x$ and $y $ and got $\cos(0)$ on the denominator which is $1$.

But I am unsure if I did that part correctly my final answer is $y-2=2(x-1)$

It is rather $1+\cos(y-2x)(\frac{dy}{dx}-2)=0$. Watch the parenthesis!
I did use the chain rule too... The derivative of $y-2x$ is $(\frac{dy}{dx}-2)$. Not $\frac{dy}{dx}(-2)=-2\frac{dy}{dx}$. Can you see the difference now?
@BabakS. Hi Babak! I will wait to see if he understands my comments first.
For the chain rule, you want to multiply $\cos(y-2x)$ by the derivative of $y-2x$. This is $\frac{dy}{dx}-2$ and not $\frac{dy}{dx}(-2)$. What do you not understand? Say $y=\sin x$ for instance. Then the derivative of $y-2x=\sin x -2x$ is $\cos x-2$, and not $(\cos x)(-2)$.

amWhy · Accepted Answer · 2013-02-21 19:12:03Z

up vote 2 down vote accepted

I think what is being suggested is that:

Differentiating $$x+\sin(y-2x)=1$$ by using the chain rule should result in:

$$1 + \cos(y - 2x)\left(\frac{dy}{dx} - 2\right) = 0$$

$$\frac{dy}{dx} = -\frac{-1 + 2\cos(y - 2x)}{\cos(y - 2x)},$$ then evaluate at point $(1,2)$

answered Feb 21 at 19:12

amWhy
54.3k770156

	The OP does not understand that the derivative of $y-2x$ is $\frac{dy}{dx}-2$, and not $\frac{dy}{dx}(-2)$. Maybe you can find better words than me. – julien Feb 21 at 19:15
	I see now I was confused between subtraction sign and the negative sign which is used to multiply. – Fernando Martinez Feb 21 at 19:18
	Yes, signs can get confusing...we all slip up, with algebra, at times. Now you need only find dy/dx evaluated at $(1, 2)$, and write the tangent line as you did in your earlier problem. – amWhy Feb 21 at 19:19
	+1 ${}{}{}{}{}{}$ – Babak S. Feb 21 at 19:20
	I see that distributive property was also used. – Fernando Martinez Feb 21 at 19:21

1 Answer