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Implicit differentiation
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Differentiating both sides of an equation with respect to \(x\) to find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) for relations not written as \(y=f(x)\), using the product and chain rules, and finding horizontal and vertical tangents and normals from the resulting derivative.
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Implicit Differentiation
AHL 5.14
Implicit differentiation is when we differentiate both sides of an equation. It is helpful when we have an equation that cannot be simplified to y=f(x).
For example:
xy2=x+y
Differentiating both sides with respect to x and using the product rule:
(x)′⋅y2+x⋅(y2)′=x′+y′
By the chain rule, we know that (y2)′=2y⋅dxdy:
1⋅y2+2xy⋅dxdy=1+dxdy
Now collecting dxdy terms:
y2−1=dxdy(1−2xy)
So
dxdy=1−2xyy2−1
Implicit Derivative: Horizontal and vertical tangents & normals
AHL 5.14
In problems involving implicit derivatives, you may be asked to solve for points where the tangent to the curve is horizontal or vertical. A horizontal tangent means dxdy=0, and a vertical tangent occurs in the case where dxdy=denominatornumerator and the denominator equals zero.
If the question asks for vertical / horizontal normals, just recall that a vertical normal means a horizontal tangent, and vice-versa.
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