Learn Differentiation Formulas with Vedantu
Functions are usually categorized under calculus in two categories, namely:
Linear functions.
Non-linear functions.
A linear function varies by its domain at a constant rate. Therefore, the overall rate of feature shift is the same as the level of function change in any situation.
Nevertheless, in the case of non-linear processes, the rate of change ranges from point to point. The variation's existence is dependent on the function's design.
The frequency of function change at a given point is known as a derivative of that function.
What is Differentiation?
Differentiation can be defined as a derivative of independent variable value and can be used to calculate features in an independent variable per unit modification.
Let,
y = f(x) be a function of x.
Then, the rate of change of “y” per unit change in “x” is given by,
\[\frac {dy} {dx}\]
If the function f(x) undergoes an infinitesimal change of h near to any point x, then the derivative of the function is depicted as
\[\lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h} \]
When a function is depicted as y = f(x),
Then the derivative is depicted by the following notations:
D(y) or D[f(x)] is called Euler's notation.
\[\frac {(dy)} {(dx)}\] is known as Leibniz’s notation.
F’(x) is known as Lagrange’s notation.
Differentiation is the method of evaluating a function's derivative at any time.
Differentiation Rules:
Some of the fundamental rules for differentiation are given below:
Sum or Difference Rule:
When the function is the sum or difference of two functions, the derivative is the sum or difference of derivative of each function, i.e.
If f(x) = u(x) ± v(x), then f’(x) = u’(x) ± v’(x)
Product Rule:
When f(x) is the sum of two u(x) and v(x) functions, it is the function derivative,
If If f(x) = u(x) x v(x),
Then f’(x) = u’(x) x v(x) + u(x) x v’(x)
Quotient Rule:
If the function f(x) is in the form of two functions u(x) / v(x), the derivative of the function can be expressed as:
If f(x) = \[\frac {u(x)} {v(x)}\],
Then f’(x) = \[\frac {u'(x)\times v(x)-u(x)\times v'(x)} {[v(x)]^2}\]
Chain Rule:
If y = f(x) = g(u),
And if u = h(x)
Then,
\[\frac {dy} {dx}\] = \[\frac {dy} {du}\] X \[\frac {du} {dx}\]
Here is a differentiation theorem collection of students so that they can turn to them to solve differential equations related problems. Higher level mathematics is one of the most important topics. The general depiction of the derivative can be expressed as \[\frac {du} {dx}\].
This list of formulas contains derivatives for constant, polynomials, trigonometric functions, logarithmic functions, hyperbolic, trigonometric inverse functions, exponential, etc. There are a number of examples and issues in classes 11 and 12 courses, which can be easily addressed by students.
Differentiation Formulas:
Differentiation is a method to find the rate of change of a function depending upon its variable or in brief the derivative of the function which is the frequency of change of function. These functions can be either linear or nonlinear depending upon the nature of slope between the points.In linear ones the slope is constant but in nonlinear ones it varies.
Let, y = f(x) be a function of x.
Then, the rate of change of “y” per unit change in “x” is given by, \[\frac {dy} {dx}\]
The differentiation formulas are those which help in solving all problems related to differentiation and its equations which may include derivatives of trigonometric functions, logarithmic functions to basic functions. They form the basis of the most important section of mathematics which is calculus. This is an easy scoring chapter. It lays the concrete foundation for the vast and advanced concepts of calculus. This concept not only helps the students to score high marks in maths but also in physics and chemistry as well.
The differentiation formulas are based on a set of rules. They are sum or difference rule, product rule, quotient rule, chain rule. Separation formulas are some of the most important differentiation formulas. Few important ones are enlisted below:
If f(x) = tan(x), then f’(x) = sec²(x)
If f(x) = cos(x) , then f’(x) = - sinx
If f(x) = sin(x) , then f’(x) = cosx
If f(x) = ln(x) , then f’(x) = \[\frac {1} {x}\]
If f(x) = ex(x) , then f’(x) = ex(x)
If f(x) = xn, then f’(x) = nxn−1
Where n is any fraction or integer.
If f(x) = k , then f’(x) = 0
Where, k is a constant.
Differentiation Formulas for Trigonometric Functions:
The definition of trigonometry is the interaction of angles and triangle faces. We have 6 major ratios here, for example, sine, cosine, tangent, cotangent, secant and cosecant. Based on these ratios, you must have learned basic trigonometric formulas. Now let's see the equations of trigonometric functions derivatives.
\[\frac {d} {dx} \](sin x) = cos x
\[\frac {d} {dx} \](cos x) = - sin x
\[\frac {d} {dx} \](tan x) = sec²x
\[\frac {d} {dx} \](cot x) = - cosec²x
\[\frac {d} {dx} \](sec x) = secx tanx
\[\frac {d} {dx} \](cosec x) = -cosecx cotx
\[\frac {d} {dx} \](sinh x) = coshx
\[\frac {d} {dx} \](cosh x) = sinhx
\[\frac {d} {dx} \](tanh x) = sech²x
\[\frac {d} {dx} \](coth x) = - cosec²x
\[\frac {d} {dx} \](sech x) = sec²x tanhx
\[\frac {d} {dx} \](cosech x) = - cosechx cotx
Differentiation Formulas for Inverse Trigonometric Functions:
Inverse equations of trigonometry are reversed proportions of trigonometry. Look at the equations of derivatives of the inverse trigonometric function.
\[\frac {d} {dx} \] \[\ [\sin^{-1} x] \] = \[\frac {1} {\sqrt{1-x^2}}\]
\[\frac {d} {dx} \] \[\ [\cos^{-1} x] \] = \[\frac {1} {\sqrt{1-x^2}}\]
\[\frac {d} {dx} \] \[\ [\tan^{-1} x] \] = \[\frac {1} {\sqrt{1+x^2}}\]
\[\frac {d} {dx} \] \[\ [\cot^{-1} x] \] = -\[\frac {1} {\sqrt{1+x^2}}\]
\[\frac {d} {dx} \] \[\ [\cos^{-1} x] \] = -\[\frac {1} {\mid x\mid\sqrt{x^2 -1}}\]
\[\frac {d} {dx} \] \[\ [\cos^{-1} x] \] = -\[\frac {1} {\mid x\mid\sqrt{x^2 -1}}\]
Differentiation Formulas List:
In all the formulas below, f’ means
\[\frac {d[f(x)]} {dx}\] = f’(x) and g’ means \[\frac {d[g(x)]} {dx}\] = g’(x).
Both f and g are the functions of x and differentiated with respect to x.
We can also represent the above equation as:
\[\frac {dy} {dx} \] = Dx y
Some of the General Differentiation Formulas are:
Power Rule:
\[\frac {d} {dx} \] xn = n xn-1
The derivative of a constant, a:
\[\frac {d} {dx}\] a = 0
Derivative of a constant multiplied with function f:
\[\frac {d} {dx}\] (a.f) = af’
Sum Rule:
\[\frac {d} {dx}\] (f ± g) = f’ ± g’
Product Rule:
\[\frac {d} {dx}\] fg = fg’ ± gf’
Quotient Rule:
\[\frac {d} {dx}\] \[\frac {f} {g}\] = \[\frac {gf'-fg'} {g^2}\]
Other Differentiation Formulas:
\[ \frac {d} {dx} \] \[ [a^x] \] = ax lna
\[ \frac{d}{dx} \] \[ [e^x] \] = ex
\[ \frac{d}{dx} log_{x}^{a} \] = 1/(Ina)x
\[\frac {d} {dx}\] (ln x) = \[\frac {1} {x} \]
Chain Rule:
\[\frac {dy} {dx} \] = \[\frac {dy} {du} \] X \[\frac {du} {dx} \]
\[\frac {dy} {dx} \] = \[\frac {dy} {dv} \] X \[\frac {dv} {du} \] X \[\frac {du} {dx} \]
Access all the differentiation formulas in the form of a pdf on our official website.
FAQs on Differentiation Formulas
1. What are the fundamental rules of differentiation?
The fundamental rules of differentiation are used to find the derivative of combined functions. The main rules include:
- Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives. d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
- Product Rule: Used for differentiating the product of two functions. d/dx [f(x)g(x)] = f(x)g'(x) + g(x)f'(x).
- Quotient Rule: Used for differentiating the ratio of two functions. d/dx [f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]².
- Chain Rule: Used for differentiating a composite function (a function within a function). If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.
2. What is the importance of differentiation formulas in real-world applications?
Differentiation is crucial for understanding rates of change. Its real-world applications include:
- Physics: To calculate instantaneous velocity and acceleration of moving objects.
- Economics: To find the maximum profit or minimum cost by analysing marginal cost and revenue functions.
- Engineering: To determine the local maxima and minima for optimizing shapes or processes.
- Medicine: To model the rate of growth of tumours or the concentration of drugs in the bloodstream over time.
3. What are the key differentiation formulas a Class 12 student must know for the 2025-26 syllabus?
As per the CBSE 2025-26 syllabus, a Class 12 student should master the derivatives of the following types of functions:
- Algebraic Functions: The power rule, d/dx(xⁿ) = nxⁿ⁻¹.
- Trigonometric Functions: Derivatives of sin(x), cos(x), tan(x), sec(x), csc(x), and cot(x).
- Inverse Trigonometric Functions: Derivatives of sin⁻¹(x), cos⁻¹(x), tan⁻¹(x), etc.
- Logarithmic Functions: The derivative of log(x) is 1/x.
- Exponential Functions: The derivative of eˣ is eˣ, and the derivative of aˣ is aˣ log(a).
4. How do you find the derivative of a function using the first principle?
Finding the derivative using the first principle involves using the definition of a derivative based on limits. The formula is: f'(x) = lim(h→0) [f(x+h) - f(x)] / h. This process calculates the instantaneous rate of change of the function f(x) by finding the slope of the tangent line to the function at point x.
5. How is the product rule different from the quotient rule in differentiation?
The main difference lies in the operation they apply to. The product rule is used when two functions are multiplied together, f(x)g(x), and its formula involves a sum. The quotient rule is used when one function is divided by another, f(x)/g(x), and its formula involves a difference and a division by the square of the denominator function. The order and sign are critical in the quotient rule, whereas they are more flexible in the product rule.
6. What is the chain rule in differentiation and why is it so important?
The chain rule is a formula to compute the derivative of a composite function, which is a function formed by combining two or more functions (e.g., sin(x²)). It's important because it allows us to differentiate complex functions that are nested within each other. The rule states you differentiate the 'outer' function while keeping the 'inner' function the same, and then multiply it by the derivative of the 'inner' function.
7. Why is the derivative of any constant function always zero?
The derivative of a function measures its rate of change. A constant function, such as f(x) = c, represents a horizontal line on a graph. Since a horizontal line has a slope of zero at every point, its rate of change is always zero. Therefore, the derivative, which geometrically represents the slope, is always zero for any constant.
8. How does differentiation relate to the slope of a curve at a single point?
Differentiation provides the precise mathematical tool to find the slope of a curve at a single point. The derivative of a function f(x) at a point x = a, denoted as f'(a), gives the exact slope of the tangent line to the curve y = f(x) at that specific point. This value represents the instantaneous rate of change of the function at that exact moment.
9. What is the core difference between differentiation and integration?
Differentiation and integration are inverse operations of each other, as stated in the Fundamental Theorem of Calculus. The core difference is their purpose:
- Differentiation is a process of finding the rate of change or the slope of a function. It breaks a problem down into smaller parts.
- Integration is a process of summing up small parts to find a whole, such as calculating the area under a curve. It 'reverses' the process of differentiation.
10. Can every continuous function be differentiated? Explain with an example.
No, not every continuous function is differentiable. A function must be smooth, without any sharp corners or cusps, to be differentiable. A classic example is the absolute value function, f(x) = |x|. This function is continuous everywhere, including at x = 0. However, it has a sharp corner at x = 0, so the slope is undefined at that point, making the function not differentiable there.
