Find the slope of the given curve at the point $(3,1)$

Question

Find the slope of the given curve at the point $(3,1)$.

$$2y\cos\left(\frac{\pi y}{x}\right)=2x^2-17y$$

How do I start? Differentiate and put the xy values in?

Answer 1

As @AndréNicolas wrote, you will have to use implicit differentiation. Then you can substitute the values for $x$ and $y$ and solve for $y'$. Here is the derivative of the left hand side:

$$\left[2y\cos\left(\frac{\pi y}{x}\right)\right]'$$ $$=[2y]'\cos\left(\frac{\pi y}{x}\right)+2y\left[\cos\left(\frac{\pi y}{x}\right)\right]'$$ $$=2y'\cos\left(\frac{\pi y}{x}\right)+2y\cdot -\sin\left(\frac{\pi y}{x}\right)\cdot \left[\frac{\pi y}{x}\right]'$$ $$=2y'\cos\left(\frac{\pi y}{x}\right)+2y\cdot -\sin\left(\frac{\pi y}{x}\right)\cdot \left(\frac{[\pi y]'\cdot x-\pi y\cdot [x]'}{x^2}\right)$$ $$=2y'\cos\left(\frac{\pi y}{x}\right)+2y\cdot -\sin\left(\frac{\pi y}{x}\right)\cdot \left(\frac{\pi y'x-\pi y}{x^2}\right)$$

You should be able to continue from there.