[Math] Find the slope of the tangent line for $y^4+3y-4x^3=5x+1$ at the point P(1,-2)

calculusderivativesimplicit-differentiationsolution-verification

Find the slope of the tangent line for $y^4+3y-4x^3=5x+1$ at the point $P(1,-2)$

Here is what I have tried. Does this look correct? Any hints or advice would be appreciated.

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EDIT 1:

Here is my new work- is this better?

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Best Answer

As indicated by somebody's comment, in the transition from your third to your fourth line, you write $\frac{d}{dx}(3y) = 3$. It's actually $3\frac{dy}{dx}$.

You did $\frac{d}{dx}(4y^3)$ correctly. I find it often helps me, when doing implicit differentiation, to remember that we are treating $y$ as a function of $x$. So something like $3y$ is actually $3f(x)$. Then implicit differentiation is just the chain rule:

$$\frac{d}{dx}(3f(x)) = 3 \cdot f'(x) \cdot \frac{d}{dx}(x) = 3f'(x)$$

where $\frac{dy}{dx}$ is what we are calling $f'(x)$.

EDIT: I believe your second work is correct.

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