derivative of
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Answer:
Derivative Rules
The Derivative tells us the slope of a function at any point.
slope examples y=3, slope=0; y=2x, slope=2
There are rules we can follow to find many derivatives.
For example:
The slope of a constant value (like 3) is always 0
The slope of a line like 2x is 2, or 3x is 3 etc
and so on.
Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark ’ means "Derivative of", and f and g are functions.
Common Functions Function
Derivative
Constant c 0
Line x 1
ax a
Square x2 2x
Square Root √x (½)x-½
Exponential ex ex
ax ln(a) ax
Logarithms ln(x) 1/x
loga(x) 1 / (x ln(a))
Trigonometry (x is in radians) sin(x) cos(x)
cos(x) −sin(x)
tan(x) sec2(x)
Inverse Trigonometry sin-1(x) 1/√(1−x2)
cos-1(x) −1/√(1−x2)
tan-1(x) 1/(1+x2)
Rules Function
Derivative
Multiplication by constant cf cf’
Power Rule xn nxn−1
Sum Rule f + g f’ + g’
Difference Rule f - g f’ − g’
Product Rule fg f g’ + f’ g
Quotient Rule f/g (f’ g − g’ f )/g2
Reciprocal Rule 1/f −f’/f2
Chain Rule
(as "Composition of Functions") f º g (f’ º g) × g’
Chain Rule (using ’ ) f(g(x)) f’(g(x))g’(x)
Chain Rule (using ddx ) dydx = dydu dudx
"The derivative of" is also written ddx
So ddxsin(x) and sin(x)’ both mean "The derivative of sin(x)"
Examples
Example: what is the derivative of sin(x) ?
From the table above it is listed as being cos(x)
It can be written as:
d/dxsin(x) = cos(x)
Or:
sin(x)’ = cos(x)
Power Rule
Example: What is d/dxx3 ?
The question is asking "what is the derivative of x3 ?"
We can use the Power Rule, where n=3:
d/dxxn = nxn−1
d/dxx3 = 3x3−1 = 3x2
(In other words the derivative of x3 is 3x2)
So it is simply this:
power rule x^3 -> 3x^2
"multiply by power
then reduce power by 1"
It can also be used in cases like this:
Example: What is d/dx(1/x) ?
1/x is also x-1
We can use the Power Rule, where n = −1:
d/dxxn = nxn−1
d/dxx−1 = −1x−1−1
= −x−2
= −1x2
So we just did this:
power rule x^-1 -> -x^-2
which simplifies to −1/x2
Multiplication by constant
Example: What is d/dx5x3 ?
the derivative of cf = cf’
the derivative of 5f = 5f’
We know (from the Power Rule):
d/dxx3 = 3x3−1 = 3x2
So:
d/dx5x3 = 5d/dxx3 = 5 × 3x2 = 15x2
Sum Rule
Example: What is the derivative of x2+x3 ?
The Sum Rule says:
the derivative of f + g = f’ + g’
So we can work out each derivative separately and then add them.
Using the Power Rule:
d/dxx2 = 2x
d/dxx3 = 3x2
And so:
the derivative of x2 + x3 = 2x + 3x2
Difference Rule
It doesn't have to be x, we can differentiate with respect to, for example, v:
Example: What is d/dv(v3−v4) ?
The Difference Rule says
the derivative of f − g = f’ − g’
So we can work out each derivative separately and then subtract them.
Using the Power Rule:
d/dvv3 = 3v2
d/dvv4 = 4v3
And so:
the derivative of v3 − v4 = 3v2 − 4v3
Sum, Difference, Constant Multiplication And Power Rules
Example: What is d/dz(5z2 + z3 − 7z4) ?
Using the Power Rule:
d/dzz2 = 2z
d/dzz3 = 3z2
d/dzz4 = 4z3
And so:
d/dz(5z2 + z3 − 7z4) = 5 × 2z + 3z2 − 7 × 4z3 = 10z + 3z2 − 28z3
Step-by-step explanation: