Algebra of Differentiation

Let f(x), g(x) and h(x) are three real valued functions are given, then

(i) Sum of two functions
\[[f(x)+g(x)]^{\prime}=f^{\prime}(x)+g^{\prime}(x)\]

(ii) Difference of two functions
\[[f(x)-g(x)]^{\prime}=f^{\prime}(x)-g^{\prime}(x)\]

(iii) Product of two or more functions

(a) \[[f(x)\cdot g(x)]^{\prime}=f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)\]

(b) \[\frac{d}{dx}[f(x)\cdot g(x)\cdot h(x)]=f(x)\cdot g(x)\cdot h^{\prime}(x)\]\[+f\left(x\right)\cdot g^{\prime}\left(x\right)\cdot h(x)+f^{\prime}\left(x\right)\cdot g\left(x\right)\cdot h\left(x\right)\]

(iv) Quotient of two functions

\[\left[\frac{f(x)}{g(x)}\right]'\] = \[\frac{f^{\prime}(x)\cdot g(x)-g^{\prime}(x)\cdot f(x)}{\left[g(x)\right]^{2}}\]

provided g(x) ≠ 0