**Differentiation Formulas:** Differentiation is one of the most important topics and perhaps the most difficult topic of Mathematics as posed by Class 11 and 12 students. There are a lot of higher-level concepts of differentiation that are taught in colleges. Therefore, it becomes important for each and every student of Science stream to have these differentiation formulas and rules at their fingertips.

In this article, we will be providing you with the list of complete differentiation formulas** **along with trigonometric formulas, formulas for logarithmic, polynomial, inverse trigonometric, and hyperbolic functions. These formulas will help you solve various problems related to differentiation.

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## Differentiation Formulas List

**Differentiation** is a process of calculating a function that represents the rate of change of one variable with respect to another. Differentiation has immense application not only in our day-to-day life but also in higher mathematics.

Let’s say y is a function of x and is expressed as y = f(x)

Then, differentiation of y with respect to x is denoted as \(\frac{dy}{dx} \).

Here, \(\frac{dy}{dx} \) represents the rate of change of y with respect to x.

It is also denoted as f'(x).Let us now look into the differentiation formulas for different functions.

### Basic Differentiation Formulas

The table below provides the derivatives of basic functions, constant, a constant multiplied with a function, power rule, sum and difference rule, product and quotient rule, etc. Differntiation formulas of basic logarithmic and polynomial functions are also provided.

a. \(\frac{d}{dx} (k)= 0\) |

b. \(\frac{d}{dx} (ku)= k\frac{du}{dx}\) |

c. \(\frac{d}{dx} (u±v)= \frac{du}{dx}±\frac{dv}{dx}\) |

d. \(\frac{d}{dx} (uv)= u\frac{dv}{dx}+v\frac{du}{dx}\) |

e. \(\frac{d}{dx} (u/v)= \frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) |

f. \(\frac{dy}{dx}.\frac{dx}{dy}= 1\) |

g. \(\frac{d}{dx} (x^n)= nx^{n-1}\) |

h. \(\frac{d}{dx} (e^x)= e^x\) |

i. \(\frac{d}{dx} (a^x)= a^x\log a\) |

j. \(\frac{d}{dx} (\log x)= \frac{1}{x}\) |

k. \(\frac{d}{dx} \displaystyle \log _{a}x= \frac{1}{x}\displaystyle \log _{a}e\) |

l. \(\frac{d^n}{dx^n} (ax+b)^n= n!a^n\) |

### Differentiation Formulas For Trigonometric Functions

Sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (cosec), and cotangent (cot) are the six commonly used trigonometric functions each of which represents the ratio of two sides of a triagnle. The derivatives of trigonometric functions are as under:

a. \(\frac{d}{dx} (\sin x)= \cos x\) |

b. \(\frac{d}{dx} (\cos x)= -\sin x\) |

c. \(\frac{d}{dx} (\tan x)= \sec^2 x\) |

d. \(\frac{d}{dx} (\cot x)= – cosec^2 x\) |

e. \(\frac{d}{dx} (\sec x)= \sec x \tan x\) |

f. \(\frac{d}{dx} (cosec x)= – cosec x \cot x\) |

g. \(\frac{d}{dx} (\sin u)= \cos u \frac{du}{dx}\) |

h. \(\frac{d}{dx} (\cos u)= -\sin u \frac{du}{dx}\) |

i. \(\frac{d}{dx} (\tan u)= \sec^2 u \frac{du}{dx}\) |

j. \(\frac{d}{dx} (\cot u)= – cosec^2 u \frac{du}{dx}\) |

k. \(\frac{d}{dx} (\sec u)= \sec u \tan u \frac{du}{dx}\) |

l. \(\frac{d}{dx} (cosec u)= – cosec u \cot u \frac{du}{dx}\) |

### Differentiation Formulas For Inverse Trigonometric Functions

Inverse trigonometric functions like (\(\sin^{-1}~ x)\) , (\(\cos^{-1}~ x)\) , and (\(\tan^{-1}~ x)\) represnts the unknown measure of an angle (of a right angled triangle) when lengths of the two sides are known. The deivatives of inverse trigonometric functions are as under:

a. \(\frac{d}{dx}(\sin^{-1}~ x)\) = \(\frac{1}{\sqrt{1-x^2}}\) |

b. \(\frac{d}{dx}(\cos^{-1}~ x)\) = -\(\frac{1}{\sqrt{1-x^2}}\) |

c. \(\frac{d}{dx}(\tan^{-1}~ x)\) = \(\frac{1}{{1+x^2}}\) |

d. \(\frac{d}{dx}(\cot^{-1}~ x)\) = -\(\frac{1}{{1+x^2}}\) |

e. \(\frac{d}{dx}(\sec^{-1}~ x)\) = \(\frac{1}{x\sqrt{x^2-1}}\) |

f. \(\frac{d}{dx}(coses^{-1}~ x)\) = -\(\frac{1}{x\sqrt{x^2-1}}\) |

g. \(\frac{d}{dx}(\sin^{-1}~ u)\) = \(\frac{1}{\sqrt{1-u^2}}\frac{du}{dx}\) |

h. \(\frac{d}{dx}(\cos^{-1}~ u)\) = -\(\frac{1}{\sqrt{1-u^2}}\frac{du}{dx}\) |

i. \(\frac{d}{dx}(\tan^{-1}~ u)\) = \(\frac{1}{{1+u^2}}\frac{du}{dx}\) |

### Differntiation Formulas For Hyperbolic Functions

The hyperbolic function of an angle is expressed as a relationship between the distances from a point on a hyperbola to the origin and to the coordinate axes. The dervatives of hyperbolic functions are as under:

a. \(\frac{d}{dx} (\sinh~ x)= \cosh x\) |

b. \(\frac{d}{dx} (\cosh~ x) = \sinh x\) |

c. \(\frac{d}{dx} (\tanh ~x)= sech^{2} x\) |

d. \(\frac{d}{dx} (\coth~ x)=-cosech^{2} x\) |

e. \(\frac{d}{dx} (sech~ x)= -sech x \tanh x\) |

f. \(\frac{d}{dx} (cosech~ x ) = -cosech x \coth x\) |

g. \(\frac{d}{dx}(\sinh^{-1} ~ x)\) = \(\frac{1}{\sqrt{x^2+1}}\) |

h. \(\frac{d}{dx}(\cosh^{-1} ~ x)\) = \(\frac{1}{\sqrt{x^2-1}}\) |

i. \(\frac{d}{dx}(\tanh^{-1} ~ x)\) = \(\frac{1}{{1-x^2}}\) |

j. \(\frac{d}{dx}(\coth^{-1} ~ x)\) = -\(\frac{1}{{1-x^2}}\) |

k. \(\frac{d}{dx}(\sec h^{-1} ~ x)\) = -\(\frac{1}{x\sqrt{1-x^2}}\) |

l. \(\frac{d}{dx}(cos h^{-1} ~ x)\) = -\(\frac{1}{x\sqrt{1+x^2}}\) |

So, now you will be quite aware of the differentiation formulas, i.e. derivatives of popular trigonometric, polynomial, inverse trigonometric, logarithmic, and hyperbolic functions.

**ASK ALL YOUR ACADEMIC QUERIES HERE**

### Important FAQs On Differentiation Formulas

You can find the important FAQs answered by our experts below:

**Q1: What are the differentiation formulae?**

**Ans:** When you calculate a function that represents the rate of change of one variable with respect to another, differentiation holds an important role there. There are a lot of deductions and derivations used which are referred to known as differentiation formulas.

**Q2: How do I memorize all the integration & differentiation formulas for trigonometry?**

**Ans:** The best way to memorize the complete complex integration and differentiation formulas is to solve questions. Start with the topics and then consistently move towards the end of the chapter. Do keep referring to these formulas whenever you get stuck on a question. You will note that you won’t need to refer to this article as it’ll get on your fingertips.

**Q3: Is there any website where I can practice differentiation formulas for free?**

**Ans:** You can practice free differential calculus questions at Embibe.

Please note that memorizing these formulas alone won’t be enough. In all likelihood, you will forget them unless you solve a sufficient number of questions to master their applications.

You can solve **differential calculus questions** for free on Embibe. These questions come with detailed explanations and solutions which will help you clarify your doubts and improve your problem-solving abilities. So, make the best use of them.

*We certainly hope that this complete list of differentiation formulas prove to be helpful for you. If you have any questions, feel free to ask in the comment section below. We will get back to you at the earliest.*

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