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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 |
|
$dfrac{d}{dx} (x) = 1$ |
|
$dfrac{d}{dx} sqrt{x} = dfrac{1}{2sqrt{x}}$ |
|
$dfrac{d}{dx}sqrt{f(x)}=dfrac{1}{2sqrt{f(x)}}dfrac{d}{dx}f(x)= dfrac{1}{sqrt{2f(x)}}f’(x)$ |
|
$dfrac{d}{dx} k cdot f(x) = kdfrac{d}{dx} f(x) = k cdot f’(x)$ |
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
$y’ = dfrac{-7}{2} x dfrac{-9}{2}$
$Rightarrow dfrac{-7}{2x^4 sqrt{x}}$