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The chain rule
applications |
Logarithmic differentiation |
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Logarithmic differentiation examples |
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Derivative of a composite exponential function |
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Use of the logarithmic differentiation |
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Derivatives of composite functions examples |
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| Logarithmic differentiation |
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The derivative of the logarithm of
a function y
= f
(x)
is called the 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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Example:
Find
the derivative of the function f
(x)
= ln (cos x).
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| Solution: |
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| Example:
Find
the
logarithmic derivative
of the function |
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| Solution:
Since the
logarithm of the given function |
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| then
differentiating both the left and the right side of the above
expression, obtained is |
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| Derivative
of a composite exponential function |
| We
use the logarithmic differentiation to find derivative of a composite
exponential function of the form, |
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| where
u
and v
are functions of the variable x
and u
> 0. |
| By
taking logarithms of both sides of the given exponential expression we
obtain, |
| ln
y
= v
ln
u. |
| Differentiating both sides of the above
equation
with respect to x |
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Use of the logarithmic differentiation |
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Derivatives of composite functions examples |
| Example:
Find
the
derivative
of the function |
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| Solution: |
 |
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| by
differentiating both sides of the above equation we get
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| or |
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| Example:
Find
the
derivative
of the function |
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| Solution: |
 |
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| by
differentiating both sides of the above equation we get
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| Example:
Find
the
derivative
of the function |
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| Solution: |
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| by
differentiating both sides of the right equation above, we get |
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| Example:
Find
the
derivative
of the function |
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| Solution: |
 |
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| by
differentiating both sides of the above equation we get
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