When you start learning calculus, one important idea you'll come across is differentiation.
Differentiation helps us figure out how a function changes when its input changes.
It’s really important for Year 12 students, especially those getting ready for their AS-Level math exams.
Let’s go over some of the basic rules of differentiation.
The power rule is super important in differentiation.
It says that if you have a function like ( f(x) = x^n ) (where ( n ) is a constant number), the derivative looks like this:
[ f'(x) = n \cdot x^{n-1} ]
For the function ( f(x) = x^3 ), using the power rule gives us:
[ f'(x) = 3 \cdot x^{2} ]
So, the derivative of ( x^3 ) is ( 3x^2 ).
The constant rule is simple.
When you differentiate a constant (like a number that doesn’t change), the derivative is always zero.
For the function ( f(x) = 5 ), the derivative would be:
[ f'(x) = 0 ]
Since constants don’t change, their rate of change is zero.
The sum rule tells us that when you find the derivative of two functions added together, you can just add their derivatives.
If ( f(x) = g(x) + h(x) ), then:
[ f'(x) = g'(x) + h'(x) ]
For ( f(x) = x^2 + 3x ), the derivative is:
[ f'(x) = 2x + 3 ]
The difference rule is like the sum rule.
It says that when you find the derivative of two functions subtracted from each other, you can subtract their derivatives.
If ( f(x) = g(x) - h(x) ), then:
[ f'(x) = g'(x) - h'(x) ]
For ( f(x) = x^2 - 4x ), the derivative would be:
[ f'(x) = 2x - 4 ]
When you have two functions multiplied together, use the product rule.
For functions ( u(x) ) and ( v(x) ), the product rule says:
[ (uv)' = u'v + uv' ]
If ( u(x) = x^2 ) and ( v(x) = \sin(x) ), then:
[ (uv)' = (2x)(\sin(x)) + (x^2)(\cos(x)) ]
The quotient rule is used for dividing two functions.
If ( f(x) = \frac{g(x)}{h(x)} ), then:
[ f'(x) = \frac{g'h - gh'}{h^2} ]
For ( f(x) = \frac{x^2}{x+1} ), applying the quotient rule gives:
[ f'(x) = \frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2} ]
Finally, the chain rule is important for functions inside other functions.
If ( y = f(g(x)) ), then the derivative is:
[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) ]
For ( y = \sin(x^2) ), we get:
[ \frac{dy}{dx} = \cos(x^2) \cdot 2x ]
These key rules—power, constant, sum, difference, product, quotient, and chain—are the building blocks of differentiation in calculus.
Learning these rules will help you tackle many math problems in Year 12 and beyond.
Happy differentiating!
When you start learning calculus, one important idea you'll come across is differentiation.
Differentiation helps us figure out how a function changes when its input changes.
It’s really important for Year 12 students, especially those getting ready for their AS-Level math exams.
Let’s go over some of the basic rules of differentiation.
The power rule is super important in differentiation.
It says that if you have a function like ( f(x) = x^n ) (where ( n ) is a constant number), the derivative looks like this:
[ f'(x) = n \cdot x^{n-1} ]
For the function ( f(x) = x^3 ), using the power rule gives us:
[ f'(x) = 3 \cdot x^{2} ]
So, the derivative of ( x^3 ) is ( 3x^2 ).
The constant rule is simple.
When you differentiate a constant (like a number that doesn’t change), the derivative is always zero.
For the function ( f(x) = 5 ), the derivative would be:
[ f'(x) = 0 ]
Since constants don’t change, their rate of change is zero.
The sum rule tells us that when you find the derivative of two functions added together, you can just add their derivatives.
If ( f(x) = g(x) + h(x) ), then:
[ f'(x) = g'(x) + h'(x) ]
For ( f(x) = x^2 + 3x ), the derivative is:
[ f'(x) = 2x + 3 ]
The difference rule is like the sum rule.
It says that when you find the derivative of two functions subtracted from each other, you can subtract their derivatives.
If ( f(x) = g(x) - h(x) ), then:
[ f'(x) = g'(x) - h'(x) ]
For ( f(x) = x^2 - 4x ), the derivative would be:
[ f'(x) = 2x - 4 ]
When you have two functions multiplied together, use the product rule.
For functions ( u(x) ) and ( v(x) ), the product rule says:
[ (uv)' = u'v + uv' ]
If ( u(x) = x^2 ) and ( v(x) = \sin(x) ), then:
[ (uv)' = (2x)(\sin(x)) + (x^2)(\cos(x)) ]
The quotient rule is used for dividing two functions.
If ( f(x) = \frac{g(x)}{h(x)} ), then:
[ f'(x) = \frac{g'h - gh'}{h^2} ]
For ( f(x) = \frac{x^2}{x+1} ), applying the quotient rule gives:
[ f'(x) = \frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2} ]
Finally, the chain rule is important for functions inside other functions.
If ( y = f(g(x)) ), then the derivative is:
[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) ]
For ( y = \sin(x^2) ), we get:
[ \frac{dy}{dx} = \cos(x^2) \cdot 2x ]
These key rules—power, constant, sum, difference, product, quotient, and chain—are the building blocks of differentiation in calculus.
Learning these rules will help you tackle many math problems in Year 12 and beyond.
Happy differentiating!