(1.)
WASSCE-FM Given that $x^2 + y^2 = 2pxy$, where
p is a constant, find
$\dfrac{dy}{dx}$
Implicit Differentiation
| Function |
Derivative |
| $x^2$ |
$
2x\dfrac{dx}{dx} \\[5ex]
2x
$
|
| $y^2$ |
$
2y\dfrac{dy}{dx}
$
|
| $2pxy$ |
$
\underline{\text{Product Rule}} \\[3ex]
Let\;\;u = 2px \hspace{3em} v = y \\[3ex]
\text{Note that p is a constant} \\[3ex]
\dfrac{du}{dx} = 2p \\[5ex]
\dfrac{dv}{dx} = \dfrac{dy}{dx} \\[5ex]
\implies \\[3ex]
2px * \dfrac{dy}{dx} + y * 2p \\[5ex]
2px\dfrac{dy}{dx} + 2py
$
|
$
x^2 + y^2 = 2pxy \\[4ex]
2x + 2y\dfrac{dy}{dx} = 2px\dfrac{dy}{dx} + 2py \\[5ex]
2y\dfrac{dy}{dx} - 2px\dfrac{dy}{dx} = 2py - 2x \\[5ex]
\dfrac{dy}{dx}(2y - 2px) = 2py - 2x \\[5ex]
\dfrac{dy}{dx} = \dfrac{2py - 2x}{2y - 2px} \\[5ex]
= \dfrac{2(py - x)}{2(y - px)} \\[5ex]
= \dfrac{py - x}{y - px}
$
(2.) Find the derivative of $x^6y + y^6x = 9\:\:wrt\:\:x$
Implicit Differentiation, Product Rule, Power Rule
$
x^6y + y^6x = 9 \\[3ex]
\dfrac{d(x^6y)}{dx} = x^6 * \dfrac{dy}{dx} + y * 6x^5 ...Product\:\:Rule \\[5ex]
\dfrac{d(x^6y)}{dx} = x^6\dfrac{dy}{dx} + 6x^5y \\[5ex]
\dfrac{d(y^6x)}{dx} = y^6 * 1 + x * 6y^5\dfrac{dy}{dx} ...Product\:\:Rule \\[5ex]
\dfrac{d(y^6x)}{dx} = y^6 + 6xy^5\dfrac{dy}{dx} \\[5ex]
\dfrac{d(9)}{dx} = 0 \\[5ex]
\rightarrow x^6\dfrac{dy}{dx} + 6x^5y + y^6 + 6xy^5\dfrac{dy}{dx} = 0 \\[5ex]
x^6\dfrac{dy}{dx} + 6xy^5\dfrac{dy}{dx} = 0 - 6x^5y - y^6 \\[5ex]
\dfrac{dy}{dx}(x^6 + 6xy^5) = -6x^5y - y^6 \\[5ex]
\dfrac{dy}{dx} = \dfrac{-6x^5y - y^6}{x^6 + 6xy^5} \\[5ex]
\dfrac{dy}{dx} = -\dfrac{(6x^5y + y^6)}{x^6 + 6xy^5}
$
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