Differentiation Formulas: Differentiation is one of the most important topics for Class 11 and 12 students. There are a lot of higherlevel concepts of differentiation that are taught in colleges. Therefore, it becomes important for each and every student of the Science stream to have these differentiation formulas and rules at their fingertips.
In this article, we will provide you with the list of complete differentiation formulas along with trigonometric formulas, formulas for logarithmic, polynomial, inverse trigonometric, and hyperbolic functions. These derivative formulas will help you solve various problems related to differentiation.
Differentiation Formulas: What Is Differentiation?
Differentiation is a process of calculating a function that represents the rate of change of one variable with respect to another. Differentiation and derivatives have immense application not only in our daytoday life but also in higher mathematics.
Differentiation Definition: Let’s say y is a function of x and is expressed as \(y=f(x)\). Then, the rate of change of “y” per unit change in “x” is given by \(\frac{dy}{dx} \).
Here, \(\frac{dy}{dx} \) is known as differentiation of y with respect to x. It is also denoted as \({f}'(x)\).
In general, if the function f(x) undergoes infinitesimal change h near to any point x, then the derivative of the function is depicted as:
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Rules Of Differentiation
There are mainly 7 types of differentiation rules that are widely used to solve problems relate to differentiation:
 Power Rule: When we need to find the derivative of an exponential function, the power rule states that:
\(\frac{d}{dx}{{x}^{n}}=n\times {{x}^{n1}}\) 
Product Rule: When \(f(x)\) is the product of two functions, \(a(x)\) and \(b(x)\), then the product rule states that:
\(\frac{d}{dx}f(x)=\frac{d}{dx}\left[ a(x)\times b(x) \right]=b(x)\times \frac{d}{dx}a(x)+a(x)\times \frac{d}{dx}b(x)\)  Quotient Rule: When \(f(x)\) is of the form \(\frac{a(x)}{b(x)}\), then the quotient rule states that:
\(\frac{d}{dx}f(x)=\frac{d}{dx}\left[ \frac{a(x)}{b(x)} \right]=\frac{b(x)\times \frac{d}{dx}a(x)a(x)\times \frac{d}{dx}b(x)}{{{\left[ b(x) \right]}^{2}}}\)  Sum or Difference Rule: When a function \(f(x)\) is the sum or difference of two functions \(a(x)\) and \(b(x)\), then the sum or difference formula states that:
\(\frac{d}{dx}f(x)=\frac{d}{dx}\left[ a(x)\pm b(x) \right]=\frac{d}{dx}a(x)\pm \frac{d}{dx}b(x)\)  Derivative of a Constant: Derivative of a constant is always zero.
Suppose \(f(x)=c\), where c is a constant. We have,
\(\frac{d}{dx}f(x)=\frac{d}{dx}(c)=0\)  Derivative of a Constant Multiplied with a Function \(f\): When we need to find out the derivative of a constant multiplied with a function, we apply this rule:
\(\frac{d}{dx}\left[ c\times f(x) \right]=c\times \frac{d}{dx}f(x)\)  Chain Rule: The chain rule of differentiation states that:
\(\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\)
Differentiation Formulas List
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. Differentiation formulas of basic logarithmic and polynomial functions are also provided.
(i) \(\frac{d}{dx} (k)= 0\) 
(ii) \(\frac{d}{dx} (ku)= k\frac{du}{dx}\) 
(iii) \(\frac{d}{dx} (u±v)= \frac{du}{dx}±\frac{dv}{dx}\) 
(iv) \(\frac{d}{dx} (uv)= u\frac{dv}{dx}+v\frac{du}{dx}\) 
(v) \(\frac{d}{dx} (u/v)= \frac{v\frac{du}{dx}u\frac{dv}{dx}}{v^2}\) 
(vi) \(\frac{dy}{dx}.\frac{dx}{dy}= 1\) 
(vii) \(\frac{d}{dx} (x^n)= nx^{n1}\) 
(viii) \(\frac{d}{dx} (e^x)= e^x\) 
(ix) \(\frac{d}{dx} (a^x)= a^x\log a\) 
(x) \(\frac{d}{dx} (\log x)= \frac{1}{x}\) 
(xi) \(\frac{d}{dx} \displaystyle \log _{a}x= \frac{1}{x}\displaystyle \log _{a}e\) 
(xii) \(\frac{d^n}{dx^n} (ax+b)^n= n!a^n\) 
Let us now look into the differentiation formulas for different types of functions.
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 triangle. The derivatives of trigonometric functions are as under:
(i) \(\frac{d}{dx} (\sin x)= \cos x\) 
(ii) \(\frac{d}{dx} (\cos x)= \sin x\) 
(iii) \(\frac{d}{dx} (\tan x)= \sec^2 x\) 
(iv) \(\frac{d}{dx} (\cot x)= – cosec^2 x\) 
(v) \(\frac{d}{dx} (\sec x)= \sec x \tan x\) 
(vi) \(\frac{d}{dx} (cosec x)= – cosec x \cot x\) 
(vii) \(\frac{d}{dx} (\sin u)= \cos u \frac{du}{dx}\) 
(viii) \(\frac{d}{dx} (\cos u)= \sin u \frac{du}{dx}\) 
(ix) \(\frac{d}{dx} (\tan u)= \sec^2 u \frac{du}{dx}\) 
(x) \(\frac{d}{dx} (\cot u)= – cosec^2 u \frac{du}{dx}\) 
(xi) \(\frac{d}{dx} (\sec u)= \sec u \tan u \frac{du}{dx}\) 
(xii) \(\frac{d}{dx} (cosec u)= – cosec u \cot u \frac{du}{dx}\) 
Differentiation Formulas For Inverse Trigonometric Functions
Inverse trigonometric functions like (\(\sin^{1}~ x)\) , (\(\cos^{1}~ x)\) , and (\(\tan^{1}~ x)\) represents the unknown measure of an angle (of a right angled triangle) when lengths of the two sides are known. The derivatives of inverse trigonometric functions are as under:
(i) \(\frac{d}{dx}(\sin^{1}~ x)\) = \(\frac{1}{\sqrt{1x^2}}\) 
(ii) \(\frac{d}{dx}(\cos^{1}~ x)\) = \(\frac{1}{\sqrt{1x^2}}\) 
(iii) \(\frac{d}{dx}(\tan^{1}~ x)\) = \(\frac{1}{{1+x^2}}\) 
(iv) \(\frac{d}{dx}(\cot^{1}~ x)\) = \(\frac{1}{{1+x^2}}\) 
(v) \(\frac{d}{dx}(\sec^{1}~ x)\) = \(\frac{1}{x\sqrt{x^21}}\) 
(vi) \(\frac{d}{dx}(coses^{1}~ x)\) = \(\frac{1}{x\sqrt{x^21}}\) 
(vii) \(\frac{d}{dx}(\sin^{1}~ u)\) = \(\frac{1}{\sqrt{1u^2}}\frac{du}{dx}\) 
(viii) \(\frac{d}{dx}(\cos^{1}~ u)\) = \(\frac{1}{\sqrt{1u^2}}\frac{du}{dx}\) 
(ix) \(\frac{d}{dx}(\tan^{1}~ u)\) = \(\frac{1}{{1+u^2}}\frac{du}{dx}\) 
Differentiation 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 derivatives of hyperbolic functions are as under:
(i) \(\frac{d}{dx} (\sinh~ x)= \cosh x\) 
(ii) \(\frac{d}{dx} (\cosh~ x) = \sinh x\) 
(iii) \(\frac{d}{dx} (\tanh ~x)= sech^{2} x\) 
(iv) \(\frac{d}{dx} (\coth~ x)=cosech^{2} x\) 
(v) \(\frac{d}{dx} (sech~ x)= sech x \tanh x\) 
(vi) \(\frac{d}{dx} (cosech~ x ) = cosech x \coth x\) 
(vii) \(\frac{d}{dx}(\sinh^{1} ~ x)\) = \(\frac{1}{\sqrt{x^2+1}}\) 
(viii) \(\frac{d}{dx}(\cosh^{1} ~ x)\) = \(\frac{1}{\sqrt{x^21}}\) 
(ix) \(\frac{d}{dx}(\tanh^{1} ~ x)\) = \(\frac{1}{{1x^2}}\) 
(x) \(\frac{d}{dx}(\coth^{1} ~ x)\) = \(\frac{1}{{1x^2}}\) 
(xi) \(\frac{d}{dx}(\sec h^{1} ~ x)\) = \(\frac{1}{x\sqrt{1x^2}}\) 
(xii) \(\frac{d}{dx}(cos h^{1} ~ x)\) = \(\frac{1}{x\sqrt{1+x^2}}\) 
So, now you are aware of the differentiation formulas, i.e. derivatives of popular trigonometric, polynomial, inverse trigonometric, logarithmic, and hyperbolic functions.
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Important FAQs On Differentiation Formulas
You can find the important FAQs answered by our experts below:
Ans: When you calculate a function that represents the rate of change of one variable with respect to another, differentiation is used and the associated formulas are differentiation formulas.
Ans: The best way to memorize the complex integration and differentiation formulas is by solving 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. With passing time, you will improve and not require the formula sheet anymore.
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. Students are advised to bookmark this page so that they can take a sneak peek at all the differentiation formulas whenever they want.
Students can make use of NCERT Solutions for Maths provided by Embibe for their exam preparation.
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We hope that this complete list of differentiation formulas helps 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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