Differentiate $y^{2} = 4 a x$ $y^2 = 4ax$ w.r.t $x$ $x$ (Where a is a constant)?

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Differentiate $y^{2} = 4 a x$ $y^2 = 4ax$ w.r.t $x$ $x$ (Where a is a constant)?

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Answer 1 · 2017-08-02T11:22:48

$\frac{d y}{d x} = \frac{2 a}{y}$ $dy/dx = (2a)/(y)$

Explanation:

When we differentiate $y$ $y$ wrt $x$ $x$ we get $\frac{d y}{d x}$ $dy/dx$ .

However, we only differentiate explicit functions of $y$ $y$ wrt $x$ $x$ . But if we apply the chain rule we can differentiate an implicit function of $y$ $y$ wrt $y$ $y$ but we must also multiply the result by $\frac{d y}{d x}$ $dy/dx$ .

Example:

$\frac{d}{d x} (y^{2}) = \frac{d}{d y} (y^{2}) \frac{d y}{d x} = 2 y \frac{d y}{d x}$ $d/dx(y^2) = d/dy(y^2)dy/dx = 2ydy/dx$

When this is done in situ it is known as implicit differentiation.

Now, we have:

$y^2=4ax \ \ \$ , the equation of a Parabola in standard form

Implicitly differentiating wrt $x$

$d/dx y^2 = d/dx 4ax$

$:. 2ydy/dx = 4a$

$:. dy/dx = (4a)/(2y)$

$:. dy/dx = (2a)/(y)$

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Differentiate $y^{2} = 4 a x$ $y^2 = 4ax$ w.r.t $x$ $x$ (Where a is a constant)?

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Explanation:

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