Linearity of differentiation
Differentiation is a linear operator; this property of the derivative which follows from the sum rule in differentiation and the constant factor rule in differentiation.Let f and g be functions. Now consider:
- <math>{d \over dx}(af(x) + bg(x))</math>
By the sum rule in differentiation, this is:
- <math>{d \over dx}(af(x)) + {d \over dx}(bg(x))</math>
By the constant factor rule in differentiation, this reduces to:
- <math>af\ '(x) + bg'(x)</math>
Hence we have:
- <math>{d \over dx}(af(x) + bg(x)) = af\ '(x) + bg'(x)</math>
Omitting the brackets[?], this is often written as:
- <math>(af + bg)' = af\ '+ bg'</math>
His footsteps went no farther down the street.
Faint and more faint the pungent perfume grew;
Temple and statue vanished from the view:
Growing sounds of discontent and deep unrest.
Tearing up the well-worn custom-bounded sod;
Into furrows, fertile furrows, rich and black.
When they widen to a field.
They will widen, they will broaden, day by day,
It will riddle all the ancient roads that lead
It will tear away the almshouse and the slum
Yes, the gardens green and sweet
In this ever-growing sound of discontent.
That the few shall yield the many larger space.
Must be given to, or taken back by Toil
Hear it breaking.html">breaking through the sod!
GO PLANT A TREE
God, what a joy it is to plant.html">plant.html">plant.html">plant.html">plant a tree.html">tree.html">tree.html">tree.html">tree.html">tree.html">tree.html">tree,
Lifting its emerald branches to the skies
Its strength and glory waxing with each spring.
To plant a tree.
Nature has many marvels; but a tree
So generous, so tender, so benign.
In pleasant converse with the winds and birds.html">birds;
To plant a tree.
Rocks are majestic; but, unlike a tree,
Of ocean billows breaking on the shore
Speaks ever of companionship and rest.
To plant a tree.
There is an oak (oh! how I love that tree)
Each day I send my blessing through the spheres
Of growing beauty.html">beauty, singing birds, and shade.
Go plant a tree.
PAIN'S PURPOSE
How blind is he who prays that God will send
Its ministry of holiness and grace.
With light and colour; and their shadows lend
The sombre background, and you lose all trace
God's majesty and beauty in each soul.
The precious pigment, pain, why then the scroll
.
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