Differentiation Laws Formulas and How to Apply Each Rule
Derivatives are the fundamental tool of calculus. The derivative of a real variable function measures the sensitivity to change the function value when its argument is changed.
Calculus Differentiation Rules
The derivative laws are the standard rules used for computing the derivative of a function in calculus.
Here, let us look into all the elementary derivative rules which are applied in calculus when finding the derivative of any function.
1) Constant Rule
According to this rule, the differentiation of any constant value is 0.
If f(x) = c where c is any constant
Then the derivate is f'(x) = 0.
2) Differentiation of a Function is Linear.
This rule provides a derivative of any function when performing addition and subtraction on the functions.
For any function f(x) and g(x) with any real numbers p and q. The derivative of this function is linear.
h(x) = a f(x) + b g(x)
The differentiation of this function is linear.
h'(x) = a f'(x) + b g'(x)
The special cases of linearity rule are:
Constant Multiple Rule
(a f(x))' = a f'(x)
Addition Rule
(f+g)' = f' + g'
Subtraction Rule
(f-g)' = f' - g'
3) Product Rule
This rule provides us with the derivative of functions when they are multiplied with each other.
For any function f(x) and g(x) the derivative of the function h(x) = f(x) g(x) with respect to x by product rule is
h'(x) = f'(x) g(x) + f(x) g'(x)
4) Chain Rule
This rule provides us with the derivative of a composite function.
For any function f(x) and g(x) the derivative of the function h(x) = f(g(x)) with respect to chain rule is
h'(x) = f'(g(x)) . g'(x)
5) Polynomial or Elementary Power Rule
The combination of the power rule with the sum and constant multiple rules allows the calculation of any polynomial's derivatives.
If f(x) = xr where r is any real number not equal to zero. Then the derivative is
f'(x) = r xr-1
When r = 1 then the function becomes a special case f(x) = x, so the derivative is f'(x)=1.
6) Quotient Rule
The quotient rule is used to find the derivative of a function when the ratio of two differentiable functions is given.
If f(x) and g(x) are two functions such that
h(x) = f(x)/ g(x)
The derivative of a function by quotient rule is
h'(x) = f'(x) g(x) + f(x) g'(x) / (g(x))2
7) The Derivative of Trigonometric Functions
The differentiation of trigonometric functions is the mathematical process for identifying the derivative or its rate of change in relation to a variable of a trigonometric function.
If f(x) = sin (x) then f'(x) = cos (x)
If f(x) = cos (x) then f'(x) = -sin (x)
If f(x) = tan (x) then f'(x) = sec2 (x)
If f(x) = sec (x) then f'(x) = sec (x) tan (x)
If f(x) = cot (x) then f'(x) = -cosec2 (x)
If f(x) = -cosec (x) then f'(x) = -cosec (x) cot (x)
8) The Derivative of an Exponential Function
An exponential function is a Mathematical function of form f (x) = ax, where x is a variable and a is a constant which is called the base of the function which is greater than 0. The differentiation of exponential function is the mathematical process of finding the rate of change in relation to a variable.
If f(x) = ax then f'(x) = ln (a) ax
If f(x) = ex then f'(x) = ex
If f(x) = ag (x) then f'(x) = ln (a) ag (x) g'(x)
If f(x) = eg (x) then f'(x) = eg (x) g'(x)
9) The Derivative of Logarithmic Functions
The inverse of exponential functions is Logarithmic functions. Logarithmic differentiation is a process used to simplify certain terms by using logarithms and their differentiation rules before effectively applying derivatives. Exponent removal, product conversion into sums and division into subtraction, which can lead to a simplified expression to derivatives, can be utilised with logarithm.
If f(x) = loga (x) then f'(x) = 1 / ln (a) x
If f(x) = ln (x) then f'(x) = 1/ x
If f(x) = loga (g(x)) then f'(x) = g'(x) / ln (a) g(x)
If f(x) = ln (g(x)) then f'(x) = g'(x) / g(x)
Problems on Calculus Differentiation Rules
1) Find the Derivative of Under Root x.
Ans: Here the given function is f(x) = \[\sqrt{x}\]
The function can be written as f(x) = (x)\[^{1/2}\]
Now differentiating the function by applying elementary power derivative rule we get
f'(x) = ½ (x)\[^{-1/2}\]
f\[^{1}\](x) = \[\frac{1}{2\sqrt{x}}\]
Therefore, the derivative of under root x is \[\frac{1}{2\sqrt{x}}\].
2) Find the Differentiation of x Cube.
Ans: Here the given function is f(x) = x3
To find the differentiation of x cube we will apply the elementary power derivative rule
f'(x) = 3x2 is the derivative of the function x3.
3) Find the Derivative of e to the Power x.
Ans: Here the given function is f(x) = ex
We will use exponential function differentiation laws to find the derivative of e power x.
So f'(x) = ex is the derivative of the function ex.
4) Find x Power x Derivative.
Ans: Here the given function is f(x) = xx
By making use of exponential function we can write f(x) = e\[^{ln(x^{x})}\] where x = e\[^{ln x}\]
Now we will use logarithmic functions differentiation laws to find the x power x derivative.
So f'(x) = (1 + ln x) xx is the derivative of the function xx.
5) Find the Derivative of e.
Ans: Here, the given function is f(x) = e
Where e is a constant function that is equal to [1+(1/n)]n. So by using the constant rule of differentiation we get
f'(x) = 0 is the derivative of the function e.
Conclusion
A derivative is the rate of change in the variable of a function.
The derivative helps us to the slope of a function at any point.
Derivatives are crucial to solving calculus and differential equation problems.
FAQs on Differentiation Laws in Calculus Explained Clearly
1. What are the laws of differentiation?
The laws of differentiation are standard rules used to find derivatives of functions quickly and accurately. These rules help differentiate algebraic, trigonometric, exponential, and logarithmic functions.
- Power Rule: d/dx (xn) = n xn−1
- Constant Rule: d/dx (c) = 0
- Constant Multiple Rule: d/dx (cf) = c f′
- Sum/Difference Rule: (f ± g)' = f′ ± g′
- Product Rule: (fg)' = f′g + fg′
- Quotient Rule: (f/g)' = (f′g − fg′)/g²
- Chain Rule: d/dx[f(g(x))] = f′(g(x)) · g′(x)
2. What is the power rule in differentiation?
The power rule states that if f(x) = xn, then its derivative is n xn−1.
- Multiply by the exponent n.
- Reduce the exponent by 1.
3. What is the constant rule in differentiation?
The constant rule states that the derivative of any constant is 0.
If f(x) = 7, then f′(x) = 0 because constants do not change with respect to x.
4. What is the product rule formula?
The product rule is used to differentiate the product of two functions and is given by (fg)' = f′g + fg′.
Steps:
- Differentiate the first function.
- Multiply by the second function.
- Add the first function multiplied by the derivative of the second.
5. What is the quotient rule in differentiation?
The quotient rule is used to differentiate a function divided by another and is given by (f/g)' = (f′g − fg′)/g².
Steps:
- Differentiate the numerator.
- Multiply by the denominator.
- Subtract numerator × derivative of denominator.
- Divide by the square of the denominator.
6. What is the chain rule in differentiation?
The chain rule is used to differentiate composite functions and is written as d/dx[f(g(x))] = f′(g(x)) · g′(x).
Example: If y = (3x + 2)4, then:
- Outer derivative: 4(3x + 2)3
- Inner derivative: 3
7. How do you differentiate a sum or difference of functions?
To differentiate a sum or difference, apply the sum rule: (f ± g)' = f′ ± g′.
Example: If f(x) = x³ + 4x, then:
- d/dx (x³) = 3x²
- d/dx (4x) = 4
8. What is the derivative of trigonometric functions?
The derivatives of basic trigonometric functions are standard results used in differentiation laws.
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec² x
9. What is the derivative of exponential and logarithmic functions?
The derivative of ex is ex, and the derivative of ln x is 1/x.
- d/dx (ex) = ex
- d/dx (ax) = ax ln a
- d/dx (ln x) = 1/x
- d/dx (loga x) = 1/(x ln a)
10. What are common mistakes when applying differentiation laws?
Common mistakes in applying differentiation laws include misusing the product, quotient, or chain rule and forgetting to simplify correctly.
- Forgetting to apply the chain rule for composite functions.
- Missing one term in the product rule.
- Incorrect sign in the quotient rule (f′g − fg′).
- Not reducing the exponent properly in the power rule.
