What are Derivatives?
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
1. What are the fundamental laws of differentiation in calculus?
The fundamental laws of differentiation are a set of rules used to find the derivative of a function. These laws simplify the process of differentiation for complex functions. The primary rules include:
- The Constant Rule: The derivative of any constant function is zero.
- The Power Rule: Used to differentiate functions of the form f(x) = xn.
- The Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives.
- The Product Rule: Used to find the derivative of a product of two functions.
- The Quotient Rule: Used to find the derivative of a function that is the ratio of two functions.
- The Chain Rule: Used to differentiate composite functions.
2. Can you explain the Product Rule and Quotient Rule with examples?
The Product and Quotient rules are essential for differentiating more complex functions.
- Product Rule: If you have two functions, u(x) and v(x), the derivative of their product is d/dx[u(x)v(x)] = u'(x)v(x) + u(x)v'(x). For example, to differentiate x²sin(x), u=x², v=sin(x). The derivative is 2x·sin(x) + x²·cos(x).
- Quotient Rule: For the ratio of two functions, u(x) and v(x), the derivative is d/dx[u(x)/v(x)] = [u'(x)v(x) - u(x)v'(x)] / [v(x)]². For example, to differentiate sin(x)/x, u=sin(x), v=x. The derivative is [cos(x)·x - sin(x)·1] / x².
3. What is the Chain Rule and why is it so important in differentiation?
The Chain Rule is a formula to compute the derivative of a composite function. If a function y = f(g(x)), then its derivative is dy/dx = f'(g(x)) · g'(x). It is crucial because it allows us to differentiate complex, 'nested' functions by breaking them down into their inner and outer parts. For instance, to differentiate sin(x²), we treat sin(u) as the outer function and u=x² as the inner function. Without the Chain Rule, differentiating most real-world functions, like those in physics and engineering, would be extremely difficult.
4. How do you differentiate exponential and logarithmic functions according to differentiation laws?
The differentiation of exponential and logarithmic functions follows specific rules as per the CBSE Class 12 syllabus for the 2025-26 session:
- Exponential Functions (ex): The function f(x) = ex is unique because its derivative is itself. So, d/dx(ex) = ex.
- General Exponential Functions (ax): The derivative of f(x) = ax is d/dx(ax) = ax log(a).
- Natural Logarithmic Functions (log x): The derivative of f(x) = log(x) (base e) is d/dx(log x) = 1/x.
5. Why is the derivative of a constant function always zero?
The derivative of a function at a point represents the instantaneous rate of change of the function at that point. A constant function, such as f(x) = 5, has the same value everywhere. This means its value does not change at all as x changes. Since there is no change, the rate of change is zero. Graphically, a constant function is a horizontal line, and the slope (which represents the derivative) of a horizontal line is always zero.
6. What is the main difference between the application of differentiation and integration?
Differentiation and integration are inverse processes, like addition and subtraction. The main difference lies in their application:
- Differentiation is used to find the rate of change of a quantity. For example, it can determine the velocity of an object from its position function or find the slope of a curve at a specific point.
- Integration is used to find the accumulation or summation of a quantity. For example, it can determine the area under a curve, the total distance travelled from a velocity function, or the volume of a solid.
7. Is it always possible to differentiate a function? What are the conditions for a function to be differentiable?
No, it is not always possible to differentiate a function at every point. For a function to be differentiable at a point, it must first be continuous at that point. However, continuity alone is not sufficient. A function is not differentiable at points where its graph has a sharp corner (like f(x) = |x| at x=0), a cusp, or a vertical tangent line. In simple terms, the graph must be 'smooth' with no abrupt changes in direction for its derivative to exist.
8. How are differentiation laws applied to find the derivatives of implicit functions?
An implicit function is one where y is not explicitly defined in terms of x, for example, x² + y² = 25. To differentiate such functions, we use a process called implicit differentiation. We differentiate both sides of the equation with respect to x, treating y as a function of x. This means whenever we differentiate a term with y, we must apply the Chain Rule and multiply by dy/dx. Finally, we algebraically solve the resulting equation for dy/dx to find the derivative.
