Question

Calc question: d^2y/dx^2 of x^2+y^2=25 at point (4,3) ?

Answer 1

Implicit differentiation to get \(\dfrac{dy}{dx}\), then implicit differentiation to find \(\dfrac{d^2y}{dx^2}\). When you're done with that, plug in \(x=4,y=3\). Do you need help with the differentiating?

Answer 2

what does implicit mean?

Answer 3

double derivation ?

Answer 4

yes i do need help

Answer 5

Suppose you're given the function \(y=x\). Finding \(\dfrac{dy}{dx}\) is fairly simple, just differentiate both sides with respect to \(x\):
\[\frac{d}{dx}[y]=\frac{d}{dx}[x]\\
\frac{dy}{dx}=1\]
But now consider the *relation* \(y^2=x\). The difference is that \(y\) is no longer a function of \(x\), but you're still capable of finding the rate of change of \(y\) with respect to \(x\). Implicit differentiation comes into play in this situation; it's essentially using the chain rule:
\[\frac{d}{dx}[y^2]=\frac{d}{dx}[x]\\
2y\frac{d}{dx}[y]=1\\
2y\frac{dy}{dx}=1\\
\frac{dy}{dx}=\frac{1}{2y}\]
If the examples don't make it clear, you can definitely learn a lot from this site: http://tutorial.math.lamar.edu/Classes/CalcI/ImplicitDiff.aspx

Answer 6

hmm, i don't see how actually

Answer 7

Let's try with your question, in that case:
\[x^2+y^2=25\]
Differentiating both sides (for the first time) gives
\[\frac{d}{dx}[x^2+y^2]=\frac{d}{dx}[25]\\
\frac{d}{dx}[x^2]+\frac{d}{dx}[y^2]=0\\
2x+2y\frac{dy}{dx}=0\\
\frac{dy}{dx}=-\frac{x}{y}\]
At the point \((4,3)\), you have
\[\frac{dy}{dx}=-\frac{4}{3}\]
Did that make sense?

Answer 8

it is d^2y/dx^2 i don't get