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Derivative of Natural Logarithmic Function: y = ln x, Lecture notes of Calculus

South Texas Vocational Technical Institute - BrownsvilleCalculus

The derivatives of various functions including trigonometric, exponential, and logarithmic functions. The focus is on the derivative of the natural logarithmic function, y = ln x. How to find the derivative using implicit differentiation and provides examples for different functions. It also includes the chain rule for finding the derivative of y = 3 ln2 5x.

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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derivatives of trigonometric, exponential & logarithmic functions
MCV4U: Calculus & Vectors
Derivative of the Natural Logarithmic Function,
y= ln x
J. Garvin
Slide 1/9
derivatives of trigonometric, exponential & logarithmic functions
Derivatives Involving ex
Recap
Determine the derivative of f(x) = 4x3
e5x.
f0(x) = e5x·12x2−4x3·5e5x
[e5x]2
=e5x(12x2−20x3)
[e5x]2
=12x2−20x3
e5x
=−
4x2(5x−3)
e5x
J. Garvin — Derivative of the Natural Logarithmic Function,y= lnx
Slide 2/9
derivatives of trigonometric, exponential & logarithmic functions
Derivative of y= ln x
Many phenomena are modelled using logarithmic functions.
A general logarithmic function, y= logbx, is related to an
exponential function.
If y= logbxthen by=x
A logarithmic function with a base of e,y= logex, is often
abbreviated y= ln xand is called the “natural logarithm”.
Note that lnex= logeex=x, and eln x=elogex=x.
J. Garvin — Derivative of the Natural Logarithmic Function,y= lnx
Slide 3/9
derivatives of trigonometric, exponential & logarithmic functions
Derivative of y= ln x
To find the derivative of y= ln x, rewrite using base eon
both sides of the equation and implicitly differentiate.
y= ln x
ey=elnx
ey=x
d
dx ey=d
dx x
eydy
dx = 1
dy
dx =1
ey
=1
x
Derivative of y= ln x
If f(x) = ln x, then f0(x) = 1
x. If y= ln x, then dy
dx =1
x.
J. Garvin — Derivative of the Natural Logarithmic Function,y= lnx
Slide 4/9
derivatives of trigonometric, exponential & logarithmic functions
Derivative of y= ln x
Example
Determine the derivative of f(x) = 8 ln x+ 5.
f0(x) = 8 1
x
=8
x
Example
Determine the derivative of f(x) = x2ln x.
f0(x) = 2xln x+x21
x
=x(2 ln x+ 1)
J. Garvin — Derivative of the Natural Logarithmic Function,y= lnx
Slide 5/9
derivatives of trigonometric, exponential & logarithmic functions
Derivative of y= ln x
Example
Determine the derivative of y= 3 ln25x.
This one uses the chain rule twice, where v= 5x,u= ln v
and y= 3u2.
dy
dx =dy
du
·du
dv
·dv
dx
= 6u·1
v
·5
= 30 ln v·1
v
=30 ln 5x
5x
=6 ln 5x
x
J. Garvin — Derivative of the Natural Logarithmic Function,y= lnx
Slide 6/9
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d e r i v a t i v e s o f t r i g o n o m e t r i c , e x p o n e n t i a l & l o g a r i t h m i c f u n c t i o n s

MCV4U: Calculus & Vectors

Derivative of the Natural Logarithmic Function,

y = ln x

J. Garvin

Slide 1/

d e r i v a t i v e s o f t r i g o n o m e t r i c , e x p o n e n t i a l & l o g a r i t h m i c f u n c t i o n s

Derivatives Involving ex

Recap Determine the derivative of f (x) =^4 x

3 e^5 x^.

f ′(x) = e

5 x (^) · 12 x (^2) − 4 x (^3) · 5 e 5 x [e^5 x^ ]^2 = e^5 x^ (12x^2 − 20 x^3 ) [e^5 x^ ]^2 =^12 x

(^2) − 20 x 3 e^5 x = − 4 x

(^2) (5x − 3) e^5 x J. Garvin — Derivative of the Natural Logarithmic Function, Slide 2/9 y = ln x

d e r i v a t i v e s o f t r i g o n o m e t r i c , e x p o n e n t i a l & l o g a r i t h m i c f u n c t i o n s

Derivative of y = ln x

Many phenomena are modelled using logarithmic functions. A general logarithmic function, y = logb x, is related to an exponential function. If y = logb x then by^ = x A logarithmic function with a base of e, y = loge x, is often abbreviated y = ln x and is called the “natural logarithm”. Note that ln ex^ = loge ex^ = x, and eln^ x^ = eloge^ x^ = x.

J. Garvin — Derivative of the Natural Logarithmic Function, Slide 3/9 y = ln x

d e r i v a t i v e s o f t r i g o n o m e t r i c , e x p o n e n t i a l & l o g a r i t h m i c f u n c t i o n s

Derivative of y = ln x

To find the derivative of y = ln x, rewrite using base e on both sides of the equation and implicitly differentiate. y = ln x ey^ = eln^ x ey^ = x d dx e y (^) = d dx x ey dydx = 1 dy dx =^

ey = (^1) x Derivative of y = ln x If f (x) = ln x, then f ′(x) = (^1) x. If y = ln x, then dydx = (^1) x. J. Garvin — Derivative of the Natural Logarithmic Function, Slide 4/9 y = ln x

d e r i v a t i v e s o f t r i g o n o m e t r i c , e x p o n e n t i a l & l o g a r i t h m i c f u n c t i o n s

Derivative of y = ln x

Example Determine the derivative of f (x) = 8 ln x + 5.

f ′(x) = 8 (^1) x = (^8) x

Example Determine the derivative of f (x) = x^2 ln x.

f ′(x) = 2x ln x + x2 1 x = x(2 ln x + 1) J. Garvin — Derivative of the Natural Logarithmic Function, Slide 5/9 y = ln x

d e r i v a t i v e s o f t r i g o n o m e t r i c , e x p o n e n t i a l & l o g a r i t h m i c f u n c t i o n s

Derivative of y = ln x

Example Determine the derivative of y = 3 ln^2 5 x. This one uses the chain rule twice, where v = 5x, u = ln v and y = 3u^2. dy dx =^ dy du ·^ du dv ·^ dv dx = 6u · (^) v^1 · 5 = 30 ln v · (^1) v = 30 ln 5 5 xx = 6 ln 5x x

J. Garvin — Derivative of the Natural Logarithmic Function, Slide 6/9 y = ln x

d e r i v a t i v e s o f t r i g o n o m e t r i c , e x p o n e n t i a l & l o g a r i t h m i c f u n c t i o n s

Derivative of y = ln x

Example Determine the equation of the tangent to y = x^2 ln 3x when x = 2e.

dy dx = 2x^ ln 3x^ +^ x 2 3 3 x = 2x ln 3x + x

When x = 2e, y = 4e^2 ln 6e and dydx

∣x=2e = 4e ln 6e + 2e.

J. Garvin — Derivative of the Natural Logarithmic Function, Slide 7/9 y = ln x

d e r i v a t i v e s o f t r i g o n o m e t r i c , e x p o n e n t i a l & l o g a r i t h m i c f u n c t i o n s

Derivative of y = ln x

Substitute x = 2e, y = 4e^2 ln 6e and m = 4e ln 6e + 2e into y = mx + b. 4 e^2 ln 6e = (4e ln 6e + 2e) · 2 e + b = 8e^2 ln 6e + 4e^2 + b b = − 4 e^2 ln 6e − 4 e^2 = − 4 e^2 (ln 6e + 1) = − 4 e^2 (ln 6 + 2) Therefore, the equation of the tangent is y = (4e ln 6e + 2e) x − 4 e^2 (ln 6 + 2).

J. Garvin — Derivative of the Natural Logarithmic Function, Slide 8/9 y = ln x

d e r i v a t i v e s o f t r i g o n o m e t r i c , e x p o n e n t i a l & l o g a r i t h m i c f u n c t i o n s

Questions?

J. Garvin — Derivative of the Natural Logarithmic Function, Slide 9/9 y = ln x