Finding the tangent line to $y= 7 \sin( \pi x+y)$ at the point $(1,0)$
Hint: Use Implicit Differentiation. We get $$\frac{dy}{dx}=7\left(\pi+\frac{dy}{dx}\right)\cos(\pi x+y).$$
Added: Now we could solve for $\frac{dy}{dx}$ and substitute. I prefer to substitute and then solve. So let $m$ be the value of $\frac{dy}{dx}$ at $(1,0)$. Substituting, we obtain $$m=7(\pi +m)(1).$$ Solve for $m$. We get $m=\frac{7\pi}{8}$.
This $m$ is the slope of our tangent line. Now I expect you can find the equation of the line that passes through $(1,0)$ and has slope $\frac{7\pi}{8}$.
For the equation of the normal, note that the slope of the normal is $\frac{8}{7\pi}$.
Related videos on Youtube
neloy
Updated on July 29, 2022Comments

neloy less than a minute
Find the lines that are tangent and normal to the curve at the given point $$y= 7 \sin( \pi x+y), \qquad (1,0)$$ The line tangent to the curve $y= 7 \sin (\pi x+y)$ at $(1,0)$ is $y=\ ?$
How would I solve this?

Admin over 8 yearsDo you know the definition of the tangent line, and its relationship with the derivative at the point?

TMM over 8 yearsNote that the constant $\pi$ is not written as "pie" but as "pi."


neloy over 8 yearsthat seems pretty different from where i ended up halfway...

neloy over 8 yearsi got dy/dx= 7(pie) cos( (pie)x + y) + 3 cos( (pie)x+y) dy/dx

André Nicolas over 8 yearsWell, continue. Set $y=0$, $x=1$, and use the fact that $\cos(\pi)=1$. So at $(1,0)$ the derivative (slope) $m$ satisfies $m=7(\pi +m)(1)$. Solve for $m$.

neloy over 8 yearsafter that i don't know what to do. in the examples it says, collect the terms with dy/dx on one side of the equation , and then solve for dy/dx. This is where i get super confused. Then it says evaluate the derivative at the point (1,0) to find the slope of the tangent. Then I'm supposed to substitute the values of the slope and the given point to find the yintercept. This is so hard :'(

André Nicolas over 8 yearsI will add some more to the post.

neloy over 8 yearsok, i think i know what to do now. gonna find yintercept. ok, put it in point slope formula it is

neloy over 8 yearsy=8/7pi + 8/7pi. ty so much for the help.

neloy over 8 yearsdarn i was wrong…….noooooooooooooo