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November 21, 201839 Insightful Publications

**Differentiation Formulas PDF:** Differentiation is one of the most important topics for Class 11 and 12 students. Therefore, every student studying in the Science stream must have a thorough knowledge of differentiation formulas and examples at their fingertips. We have provided a list of differentiation formulas for students’ reference so that they can use it to solve problems based on differential equations.

In this article, we have provided 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 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 day-to-day 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:

\(\underset{h\to \infty }{\mathop{\lim }}\,\frac{f(x+h)-f(x)}{h}\)

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:

\(\frac{d}{dx}{{x}^{n}}=n\times {{x}^{n-1}}\)-
**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:

\(\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\)

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

(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.

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

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

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

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

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

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

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

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

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

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

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

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

(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. You can download Differentiation Formulas cheat sheet and Pdf on Embibe.

Download – Differentiation Formulas PDF |

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Mensuration Formulas | HCF |

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 is used and the associated formulas are differentiation formulas. |

Q2: How do I memorize all the integration and differentiation formulas for trigonometry?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. |

Q3: Is there any website where I can practice differentiation formulas for free?Ans: You can practice free differential calculus questions at Embibe. |

Q4. What are the common formulas of differentiation?Ans: The common formulas of differentiation include: Derivatives of basic functions, Derivatives of Logarithmic and Exponential functions, Derivatives of Trigonometric functions, Derivatives of Inverse trigonometric functions, Differentiation rules. |

Q5. What are some of the basic rules of differentiation?Ans: Some of the basic rules of differentiation are:Power Rule: (d/dx) (x^n ) = nx^{n-1} Product Rule: (d/dx) (fg)= fg’ + gf’ Quotient Rule: (d/dx) (f/g) = [(gf’ – fg’)/g^2] Sum Rule: (d/dx) (f ± g) = f’ ± g’ |

Q6. What are some of the commonly used derivatives of trigonometric functions?Ans: The commonly used derivatives of six trigonometric functions are:(d/dx) sin x = cos x (d/dx) cos x = -sin x (d/dx) tan x = sec^2 x (d/dx) sec x = sec x tan x (d/dx) cot x = -cosec^2 x (d/dx) cosec x = -cosec x cot x |

Q7. What is d/dx?Ans: d/dx is the general representation of the derivative function. d/dx denotes the differentiation with respect to the variable x. |

Q8. What is a UV formula?Ans: (d/dx)(uv) = v(du/dx) + u(dv/dx)This formula is used to find the derivative of the product of two functions. |

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