Derivative Formulas

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All Derivative Formulas

Derivatives are one of the fundamental tools that are widely used to solve different problems on calculus and differential equations. It is one of the important topics of calculus. The questions based on derivatives are not only asked in school, but also in competitive exams like JEE Main, JEE advance, etc. Even, the concepts of derivatives are used to solve different real-life problems such as calculating the profit and loss of business, checking the variation in temperature, determining the speed or distance covered such as kilometres per hour, miles per hour, etc.


Hence, the students need to be well-versed with the derivative formulas and derivative rules. Here, you will find a list of all derivative formulas, along with derivative rules that will be helpful for you to solve different problems on differentiation.


Derivative in Maths

In Mathematics, the derivative is a method to show the instantaneous rate of change, that is the amount by which a function changes at a given point of time. The derivatives are often represented as $\dfrac{dy}{dx}$ (spelt as $dy$ over $dx$, meaning the difference in $y$ is divided by difference in $x$). The $d$ in $\dfrac{dy}{dx}$ is not a variable and cannot be cancelled. Another common representation of derivative is $f’(x)$ - meaning the derivative of a function $f$ at point $x$.


Derivative Definition

Let $f(y)$ be a function whose domain includes an open interval about some point $y_0$. Then, as the function $f(y)$ is considered to be differentiable at $y_0$ and derivative of $f(y)$ at $y_0$ is given as:

$f'(y_0)= \underset{y \to 0}{lim} \dfrac{\Delta x}{\Delta y} = \underset{y \to 0}{lim} \dfrac{f(y_0 + \Delta y) - f(y_0)}{\Delta y}$

List of All Derivative Formulas

Find the list of all derivative formulas:


General Derivative Formulas

Functions

$\dfrac{d}{dx} (k) = 0$, where $k$ is any constant

$\dfrac{d}{dx} (x) = 1$

$\dfrac{d}{dx} \sqrt{x} = \dfrac{1}{2\sqrt{x}}$

$\dfrac{d}{dx}\sqrt{f(x)}=\dfrac{1}{2\sqrt{f(x)}}\dfrac{d}{dx}f(x)= \dfrac{1}{\sqrt{2f(x)}}f’(x)$

$\dfrac{d}{dx} k \cdot f(x) = k\dfrac{d}{dx} f(x) = k \cdot f’(x)$


Derivatives of Logarithmic Function

Functions

Derivatives

$\dfrac{d}{dx} In x$

$\dfrac{1}{x}$

$\dfrac{d}{dx} log_a x$

$\dfrac{1}{x In a}$

$\dfrac{d}{dx} In f(x)$

$\dfrac{f'(x)}{f (x)}$

$\dfrac{d}{dx} log_a f(x)$

$\dfrac{1}{f(x) In a} \dfrac{d}{dx}f(x)$


Derivatives of Exponential Function Formulas

Functions

Derivatives 

$\dfrac{d}{dx}e^x$

$e^x$

$\dfrac{d}{dx}a^x$

$a^x In a$

$\dfrac{d}{dx} x^x$

$x^x(1+In x)$

$\dfrac{d}{dx}e^{f(x)}$

$e^{f(x)} \dfrac{d}{dx} f(x)$

$\dfrac{d}{dx}a^{f(x)}$

$a^{f(x)}In af(x)$


Derivatives of Trigonometric Functions Formulas

Functions

Derivatives

$\dfrac{d}{dx} (sin(x))$

$cos (x)$

$\dfrac{d}{dx} (cos(x))$

$-sin (x)$

$\dfrac{d}{dx} (tan(x))$

$sec^2 (x)$

$\dfrac{d}{dx} (sec(x))$

$sec (x) tan (x)$

$\dfrac{d}{dx} (csc(x))$

$-csc (x) cot x$

$\dfrac{d}{dx} (cot(x))$

$-csc^2(x)$


Derivatives of Inverse Trigonometric Functions

Function

Derivative

Restrictions

$\dfrac{d}{dx} (sin^{-1} x)$

$\dfrac{1}{1 - x^2}$ 

$-1 < x < 1$

$\dfrac{d}{dx}(cos^{-1} x)$

$\dfrac{-1}{1 - x^2}$

$-1 < x < 1$

$\dfrac{d}{dx}(tan^{-1} x)$

$\dfrac{1}{1 + x^2}$


$\dfrac{d}{dx}(sec^{-1} x)$

$\dfrac{1}{|x| \sqrt{x^2 - 1}}$

$|x| > 1$

$\dfrac{d}{dx}(csc^{-1} x)$

$\dfrac{-1}{|x| \sqrt{x^2 - 1}}$

$|x| > 1$

$\dfrac{d}{dx}(cot^{-1} x)$

$\dfrac{-1}{1 + x^2}$



Derivatives of Hyperbolic Functions

Function

Derivative

$\dfrac{d}{dx} (sinh x)$

$cosh x$

$\dfrac{d}{dx}(cosh x)$

$sinh x$

$\dfrac{d}{dx}(tanh x)$

$sech^2 x$

$\dfrac{d}{dx}(csch x)$

$-cschx cothx$

$\dfrac{d}{dx}(sech x)$

$-sechx tanhx$

$\dfrac{d}{dx}(coth x)$

$-csch^2 x$


Derivatives of Inverse Hyperbolic Function

Inverse Function

Derivative

Restrictions

$y sinh^{-1}x$

$y' = \dfrac{1}{1+x^2}$


$y cosh^{-1}x$

$y' = \dfrac{1}{x^2-1}$


$y tanh^{-1}x$

$y' = \dfrac{1}{1-x^2}$

$|x|  < 1$

$y csc h^{-1}x$

$y'= \dfrac{-1}{\sqrt[x]{1+x^2}}$

$x > 0$

$y sech^{-1}x$

$y' = \dfrac{-1}{\sqrt[x]{1-x^2}}$

$0 < x < 1$

$y coth^{-1}x$

$y' = \dfrac{1}{x^2-1}$

$|x| > 1$


Derivative Rules

Constant Rule

Let $k$ be a constant, then $\dfrac{d}{dx} (n) = 0$.


The derivative of constant always equals to $0$


Power Rule

If $n$ is any real number, then

$\dfrac{d}{dx} (x^n) = nx^{n-1}$

If $n$ is any positive integer, then

$\dfrac{d}{dx}(x^n) = nx^{n-1}$


Constant Multiple Rule

If $k$ is a constant, and $f$ is differentiable, then

$\dfrac{d}{dx} [k f(x)]= k \dfrac{d}{dx} f(x)$


Sum and Differentiable Rule

If $a$ and $b$ both are differentiable, then

$\dfrac{d}{dx}(a (x)+b (x)) = \dfrac{d}{dx} a(x) + \dfrac{d}{dx} b(x)$

And,

$\dfrac{d}{dx}(a (x)+b (x))= \dfrac{d}{dx} a(x) + \dfrac{d}{dx} b(x)$


Product Rule

If $a$ and $b$ both are differentiable then,

$\dfrac{d}{dx}[a (x) \cdot b(x)] = a(x) \dfrac{d}{dx}[b (x)] +b(x) \dfrac{d}{dx}[a (x)]$


Quotient Rule

If $a$ and $b$ both are differentiable then,

$\dfrac{d}{dx}[a(x)b(x)] = \dfrac{b(x) \dfrac{d}{dx}a(x) - a(x)\dfrac{d}{dx}b(x)}{(b(x))^2}$


Solved Problems

1. Find the derivative of the exponential function $f(x) =2 \cdot 3^x$

Sol: $f'(x) =2(3^x)' = 2 \cdot 3^x In 3$

$\Rightarrow = 2 (In)3 \cdot 3^x$

Hence, the derivative of the exponential function $f(x) =2 \cdot 3^x$ is $2(In)3 \cdot 3^x$


2. Find the derivative of the logarithmic function $f(x) = In x + x$.

Sol: $f(x) = (In x)' + (x)' = \dfrac{1}{x}+ 1$

Hence, the derivative of the exponential function $f(x) = In x + x$ is $\dfrac{1}{x}+ 1$


3. Find the derivative of the function $y = \dfrac{1}{x^2 \sqrt{x^3}}$

Sol: As $y = \dfrac{1}{x^2 \sqrt{x^3}}= x^{\frac{-7}{2}}$

Accordingly, we have

$y' = \dfrac{-7}{2} x \dfrac{-9}{2}$

$\Rightarrow \dfrac{-7}{2x^4 \sqrt{x}}$

FAQ (Frequently Asked Questions)

Q1. What are the Applications of Derivative Formulas?

Ans. Derivative formulas are used to find the following factors:

  • Rate of change of a quantity.

  • Maxima and minima of a functions.

  • Equations of tangents and normal for a curve to the point.

  • To determine the interval in which function is increasing or decreasing.

  • To find the turning point of a curve.

  • To find the approximate value of a given quantity.

Q2. What are Derivative Rules?

Ans. Derivative rules are the rules that are used to find the derivative of a function in calculus.

Q3. Who Introduced Derivatives?

Ans. Two different notations such as Leibniz notation, and Lagrange notation are commonly used in derivatives, one is derived by Gottfried Wilhelm Leibniz and the other by Joseph Louis Lagrange.