Differentiation Formulas List
Basic Formulas of Derivatives
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\LARGE \begin{array}{l} \frac{d(f(x))}{dx} = f'(x)\end{array}
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\LARGE\begin{array}{l}\frac{d(g(x))}{dx}= g'(x) \end{array}
1. Power Rule: (d/dx) (\LARGE x^n ) = \LARGE nx^{n-1}
2. Derivative of a constant, a: \LARGE (d/dx) (a) = 0
3. Derivative of a constant multiplied with function ,
f: \LARGE (d/dx) (a. f) = af’
4. Sum Rule: \LARGE (d/dx) (f ± g) = f’ ± g’
5. Product Rule: \LARGE (d/dx) (fg)= fg’ + gf’
6. Quotient Rule:
\LARGE \begin{array}{l}\frac{d}{dx}(\frac{f}{g})= \frac{gf’ – fg’}{g^2}\end{array}
Differentiation Formulas for Trigonometric Functions
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\begin{array}{l}\frac{d}{dx} (sin~ x)= cos\ x\end{array}
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\begin{array}{l}\frac{d}{dx} (cos~ x)= – sin\ x\end{array}
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\begin{array}{l}\frac{d}{dx} (tan ~x)= sec^{2} x\end{array}
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\begin{array}{l}\frac{d}{dx} (cot~ x = -cosec^{2} x\end{array}
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\begin{array}{l}\frac{d}{dx} (sec~ x) = sec\ x\ tan\ x\end{array}
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\begin{array}{l}\frac{d}{dx} (cosec ~x)= -cosec\ x\ cot\ x\end{array}
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\begin{array}{l}\frac{d}{dx} (sinh~ x)= cosh\ x\end{array}
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\begin{array}{l}\frac{d}{dx} (cosh~ x) = sinh\ x\end{array}
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\begin{array}{l}\frac{d}{dx} (tanh ~x)= sech^{2} x\end{array}
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\begin{array}{l}\frac{d}{dx} (coth~ x)=-cosech^{2} x\end{array}
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\begin{array}{l}\frac{d}{dx} (sech~ x)= -sech\ x\ tanh\ x\end{array}
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\begin{array}{l}\frac{d}{dx} (cosech~ x ) = -cosech\ x\ coth\ x\end{array}
Differentiation Formulas for Inverse Trigonometric Functions
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\begin{array}{l}\frac{d}{dx}(sin^{-1}~ x)=\frac{1}{\sqrt{1 – x^2}}\end{array}
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\begin{array}{l}\frac{d}{dx}(cos^{-1}~ x) = -\frac{1}{\sqrt{1 – x^2}}\end{array}
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\begin{array}{l}\frac{d}{dx}(tan^{-1}~ x) = \frac{1}{1 + x^2}\end{array}
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\begin{array}{l}\frac{d}{dx}(cot^{-1}~ x) = -\frac{1}{1 + x^2}\end{array}
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\begin{array}{l}\frac{d}{dx}(sec^{-1} ~x) = \frac{1}{|x|\sqrt{x^2 – 1}}\end{array}
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\begin{array}{l}\frac{d}{dx}(cosec^{-1}~x) = -\frac{1}{|x|\sqrt{x^2 – 1}}\end{array}
Other Differentiation Formulas
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\begin{array}{l}\frac{d}{dx}(a^{x}) = a^{x} ln a\end{array}
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\begin{array}{l}\frac{d}{dx}(e^{x}) = e^{x}\end{array}
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\begin{array}{l}\frac{d}{dx}(log_a~ x) = \frac{1}{(ln~ a)x}\end{array}
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\begin{array}{l}\frac{d}{dx}(ln~ x) = 1/x\end{array}
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Chain Rule:
\LARGE \begin{array}{l}\frac{dy}{dx}= \frac{dy}{du}\times \frac{du}{dx}= \frac{dy}{dv}\times \frac{dv}{du}\times \frac{du}{dx}\end{array}
What are the Differentiation of tanx?
\LARGE \begin{array}{l}\frac{d}{dx} (tanh ~x)= sech^{2} x\end{array}
What are the Differentiation of cot x?
\LARGE \begin{array}{l}\frac{d}{dx} (cosh~ x) = sinh\ x\end{array}
What are the Differentiation of log x?
\LARGE f(x) = \log_a{x}
\LARGE f'(x) = \dfrac{1}{x \cdot \ln{a}}
\LARGE f(x) = \log_a{g(x)}
\LARGE f'(x) = \dfrac{g'(x)}{g(x) \cdot \ln{a}}
What are the Differentiation of sinx?
\LARGE \begin{array}{l}\frac{d}{dx} (sin~ x)= cos\ x\end{array}
What are the differentiation of tan inverse x?
\LARGE \begin{array}{l}\frac{d}{dx}(tan^{-1}~ x) = \frac{1}{1 + x^2}\end{array}
