Download Derivative of the Natural Logarithmic Function, y=ln(x)

derivatives of trigonometric, exponential & logarithmic functions
derivatives of trigonometric, exponential & logarithmic functions
Derivatives Involving e x
MCV4U: Calculus & Vectors
Recap
Determine the derivative of f (x) =
Derivative of the Natural Logarithmic Function,
y = ln x
4x 3
.
e 5x
e 5x · 12x 2 − 4x 3 · 5e 5x
[e 5x ]2
5x
2
e (12x − 20x 3 )
=
[e 5x ]2
12x 2 − 20x 3
=
e 5x
4x 2 (5x − 3)
=−
e 5x
f 0 (x) =
J. Garvin
J. Garvin — Derivative of the Natural Logarithmic Function, y = ln x
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derivatives of trigonometric, exponential & logarithmic functions
derivatives of trigonometric, exponential & logarithmic functions
Derivative of y = ln x
Derivative of y = ln x
Many phenomena are modelled using logarithmic functions.
To find the derivative of y = ln x, rewrite using base e on
both sides of the equation and implicitly differentiate.
A general logarithmic function, y = logb x, is related to an
exponential function.
y = ln x
e y = e ln x
y
If y = logb x then b = x
ey = x
A logarithmic function with a base of e, y = loge x, is often
abbreviated y = ln x and is called the “natural logarithm”.
d y
dx e
e y dy
dx
Note that ln e x = loge e x = x, and e ln x = e loge x = x.
dy
dx
=
d
dx x
=1
1
= y
e
= x1
Derivative of y = ln x
If f (x) = ln x, then f 0 (x) = x1 . If y = ln x, then
J. Garvin — Derivative of the Natural Logarithmic Function, y = ln x
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derivatives of trigonometric, exponential & logarithmic functions
Derivative of y = ln x
Derivative of y = ln x
Example
Determine the derivative of f (x) = 8 ln x + 5.
Determine the derivative of y = 3 ln2 5x.
=
8
x
This one uses the chain rule twice, where v = 5x, u = ln v
and y = 3u 2 .
dy
dx
=
=
Example
=
Determine the derivative of f (x) = x 2 ln x.
f 0 (x) = 2x ln x + x 2 x1
dy
du
du dv
dv · dx
6u · v1 · 5
30 ln v · v1
·
30 ln 5x
5x
6 ln 5x
=
x
=
= x(2 ln x + 1)
J. Garvin — Derivative of the Natural Logarithmic Function, y = ln x
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= x1 .
derivatives of trigonometric, exponential & logarithmic functions
Example
f 0 (x) = 8 x1
dy
dx
J. Garvin — Derivative of the Natural Logarithmic Function, y = ln x
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J. Garvin — Derivative of the Natural Logarithmic Function, y = ln x
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derivatives of trigonometric, exponential & logarithmic functions
Derivative of y = ln x
derivatives of trigonometric, exponential & logarithmic functions
Derivative of y = ln x
Example
Determine the equation of the tangent to y =
x = 2e.
x 2 ln 3x
when
Substitute x = 2e, y = 4e 2 ln 6e and m = 4e ln 6e + 2e into
y = mx + b.
4e 2 ln 6e = (4e ln 6e + 2e) · 2e + b
= 8e 2 ln 6e + 4e 2 + b
dy
dx
b = −4e 2 ln 6e − 4e 2
3
= 2x ln 3x + x 2 3x
= 2x ln 3x + x
When x = 2e, y = 4e 2 ln 6e and dy
dx x=2e
= −4e 2 (ln 6e + 1)
= 4e ln 6e + 2e.
J. Garvin — Derivative of the Natural Logarithmic Function, y = ln x
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derivatives of trigonometric, exponential & logarithmic functions
Questions?
J. Garvin — Derivative of the Natural Logarithmic Function, y = ln x
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= −4e 2 (ln 6 + 2)
Therefore, the equation of the tangent is
y = (4e ln 6e + 2e) x − 4e 2 (ln 6 + 2).
J. Garvin — Derivative of the Natural Logarithmic Function, y = ln x
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