Find the slope of the tangent line to the curve defined by $7x^4 - 8xy - 6y^3 = 322$ at the point $(2, -3)$

Question

Find the slope of the tangent line to the curve defined by

$$ 7x^4 - 8xy - 6y^3 = 322$$ at the pont $(2, -3)$

I'm having a tough time using implicit differentiation and chain rule with all the forms of y in the problem, help is greatly appreciated

Answer 1

Hint: $$\frac{d}{dx} xy = y+x \frac{dy}{dx}$$

If the question does not require you to find the simplest form of $\frac{dy}{dx}$ , then just simply put $(x,y)=(2,-3)$ after taking the first derivative on both side, after that you will find out what does the slope is.

Answer 2

Key takeaway is that for a function $y=y(x)$ then, upon considering

$z(x) = xy$

One finds $z'(x)$ by first the product rule $$\frac{d}{dx} a(x) b(x) = a \frac{d b}{dx}+ \frac{d a}{dx}b$$ and via 'implicitly differentiating $y$ where $$\frac{d}{dx}f(y) = \frac{d}{dy}f(y) \times \frac{dy}{dx}$$ Thus, in my example above, one has \begin{align} \frac{d}{dx}xy &= \frac{dx}{dx}y+x \frac{dy}{dx} \\ &= y+x \frac{dy}{dx} \end{align}

Applying this to the second term in your example above yields \begin{align} \frac{d}{dx}(-8xy) &=\frac{d}{dx}(-8)xy-(8)\frac{d}{dx}(x)y-(8)x\frac{d}{dx}(y) \\ &=-8\frac{d}{dx}(x)y-8x\frac{d}{dx}(y) \\ &= -(8)\frac{dx}{dx}y-8x\frac{dy}{dx} \\ &= -8y-8x\frac{dy}{dx} \\ &= -8y-8xy' \\ \end{align}

Now, clearly, the trickest part of your problem lies in the non-lienar part $y^3$ but the application of the above results, along with the 'power rule' (see below) you are aware of already should give you enough momemtum to go forward.

Power Rule: For $y=y(x)=x^n, \frac{dy}{dx} = n x ^{n-1}$

Post comments edit

It is clear the term in $y^3$ is a stumbling block.

Let me walk you through it, using some of the concepts above. Set $z(x) = 6y^3 = 6y(x)^3$

Now, \begin{align} \frac{d}{dx} 6y^3&= 6\frac{d}{dy} \times \frac{dy}{dx} y^3 \\ &= 6\frac{d}{dy} y^3 \times \frac{dy}{dx} \\ &= 18y'y^2 \end{align}

Hence

$$28x^3 - 8y - 8xy'-18y'y^2 = 0$$

Or, alternatively $$y' = \frac{28 x^3-8y}{8x+18y^2}$$

Now, insert the points $(x, y)=(2, -3)$ and see if this agrees with the answer required.

Further Edit

The Chain rule states For $z = z(y(x))$; $$\frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}$$

Here, your y' is indeterminate, ie, this is the form we include in the above 'graident function'. The other part can be calculated explicitly, viz $$z=y^2 = y(x)^3 \, \implies \,\frac{dz}{dx} = \frac{d}{dy}(y^3) \frac{dy}{dx} = 3y^2 \frac{dy}{dx} = 3y^2y'$$

Does this help?