Rules of Differentiation

Differentiation is one of the most basic concepts in calculus. It helps us find the rate of change of a function at a given point. The beauty of differentiation lies in its practicality - it is used in many fields such as physics, engineering, economics, statistics, and more. This lesson will cover important rules of differentiation, supporting each with explanations, examples, and visual representations.

Power law

The power rule is the simplest and most commonly used rule in differentiation. It states that for any function of the form ( f(x) = ax^n ), where ( a ) is a constant and ( n ) is a positive integer, the derivative ( f'(x) ) is given by:

f'(x) = anx^{n-1}

Let's look at an example:

Example: Find the derivative of ( f(x) = 3x^4 ).

Solution:

f(x) = 3x^4 f'(x) = 3 * 4 * x^{4-1} f'(x) = 12x^3

Continuation rule

The constant rule states that the derivative of a constant function is zero. If ( f(x) = c ), where ( c ) is a constant, then:

f'(x) = 0

Example: Find the derivative of ( f(x) = 7 ).

Solution:

f(x) = 7 f'(x) = 0

Sum rules

The sum rule states that the derivative of a sum of functions is the sum of their respective derivatives. If we have two functions ( f(x) ) and ( g(x) ), then the derivative of their sum is:

(f + g)'(x) = f'(x) + g'(x)

Example: If ( f(x) = x^2 + 3x ), then find ( f'(x) ).

Solution:

f(x) = x^2 + 3x f'(x) = (x^2)' + (3x)' f'(x) = 2x + 3

Difference rule

The difference rule is similar to the sum rule but it applies to the difference of functions. If we have two functions ( f(x) ) and ( g(x) ), then the derivative of their difference is:

(f - g)'(x) = f'(x) - g'(x)

Example: If ( f(x) = 4x - 5x^3 ), then find ( f'(x) ).

Solution:

f(x) = 4x - 5x^3 f'(x) = (4x)' - (5x^3)' f'(x) = 4 - 15x^2

Product rule

The multiplication rule states that the derivative of the product of two functions is obtained by multiplying the derivative of the first function by the second function and multiplying the first function by the derivative of the second function. If ( f(x) ) and ( g(x) ) are two differentiable functions, then:

(f cdot g)'(x) = f'(x) cdot g(x) + f(x) cdot g'(x)

Example: If ( f(x) = x^2 ) and ( g(x) = sin(x) ), then find ( (f cdot g)'(x) ).

Solution:

f(x) = x^2, g(x) = sin(x) f'(x) = 2x, g'(x) = cos(x) (f cdot g)'(x) = (x^2)' cdot sin(x) + x^2 cdot (sin(x))' (f cdot g)'(x) = 2x cdot sin(x) + x^2 cdot cos(x)

Quotient rule

The quotient rule is used to differentiate functions that are divided by each other. If ( f(x) ) and ( g(x) ) are two differentiable functions, then the derivative of their quotient is:

(f/g)'(x) = (g(x) cdot f'(x) - f(x) cdot g'(x)) / (g(x))^2

Example: If ( f(x) = x^2 ) and ( g(x) = e^x ), then find ( (f / g)'(x) ).

Solution:

f(x) = x^2, g(x) = e^x f'(x) = 2x, g'(x) = e^x (f / g)'(x) = (e^x cdot 2x - x^2 cdot e^x) / (e^x)^2 (f / g)'(x) = (2xe^x - x^2e^x) / e^{2x}

Chain rule

The chain rule is a formula to calculate the derivative of a composite function. If one variable ( u ) is a function of ( x ), i.e., ( u = g(x) ) and the other variable ( y ) is a function of ( u ), i.e., ( y = f(u) ), then the derivative of ( y ) with respect to ( x ) is:

dy/dx = dy/du cdot du/dx

Example: If ( y = (3x + 2)^5 ), then find ( dy/dx ).

Solution:

Let u = 3x + 2 y = u^5 dy/du = 5u^4 du/dx = 3 dy/dx = 5(3x + 2)^4 cdot 3 dy/dx = 15(3x + 2)^4

Conclusion

The rules of differentiation are powerful tools that make differentiation tasks systematically easier. With practice, these rules enable a person to efficiently solve complex problems in calculus. Each rule addresses different types of tasks, whether they are products, quotients, or compounds. Understanding how and when to apply each rule is the key to mastering differentiation.

Building a thorough understanding of these fundamental rules will help you greatly as you move on to more complex calculus topics like integration or partial derivatives. Keep practicing problems, and soon you'll find that applying these rules becomes second nature.

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