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The chain rule
applications
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Implicit
differentiation
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Let
given a function F
= [ y (x)] n,
to
differentiate F
we use the power rule and the chain rule,
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| which,
for example, for n =
2 gives |
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The
same method we use to differentiate an equation of a curve given in the
implicit form
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| F
(x,
y) = c or
F
(x,
y) = 0. |
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So,
the chain rule applied to differentiate the above equation with respect
to x
gives
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Meaning,
differentiate the equation with respect to both variables at the same
time such that, when differentiating
with respect to x,
consider y
constant,
while when differentiating
with respect to y,
consider x constant
and multiply by y'. |
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Implicit differentiation examples
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| Example:
Calculate
the derivative y' of the
equation of the curve x2
+ y2 -
2axy
= 0.
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Solution:
We
use the above method of implicit differentiation to find y'
of the
equation,
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Example:
Let
determine the equation of the line tangent to the ellipse b2x2
+ a2y2 = a2b2
with the center at the origin, at the point (x1,
y1).
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| Solution:
Differentiating
the equation of the ellipse we get,
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so
the slope |
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of the line
tangent to the ellipse at the point (x1,
y1). |
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Therefore,
the equation of the line tangent to the ellipse
at the point (x1,
y1)
is
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Generalized
power rule
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| Example:
Find
the derivative of the function |
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| Solution:
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| Example:
Find
the derivative of the function |
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| Solution:
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Logarithmic differentiation
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The derivative of the logarithm of
a function y
= f (x)
is called logarithmic derivative of the
function, thus
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Therefore, the logarithmic derivative
is the derivative of the logarithm of a given function.
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Logarithmic differentiation examples
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Example:
Find
the derivative of the function f(x)
= ln (sin x).
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| Solution:
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