# Rules of Differentiation in Calculus

 

In what follows $f$, $g$, and $h$ are functions of the variable x. k is a constant.

 Definition of the Derivative of a Function The derivative $\dfrac{dy}{dx}$ of function $y=f(x)$ is defined as $\displaystyle \dfrac{dy}{dx}=\lim_{h\to 0} \dfrac{f(x+h)-f(x)}{h}$ Other notations of the derivative are: $\dfrac{df}{dx}$ , $y'$ or $f'$. Basic Rules $\dfrac{d k}{d x}=0$ , for any constant $k$ $\dfrac{d }{d x}(k f)=k \dfrac{d f}{d x}$ $\dfrac{d }{d x} x^n = n x^{n-1}$ $\dfrac{d }{d x}[f(x)+g(x)] =\dfrac{d }{dx}f(x)+\dfrac{d}{dx}g(x)$ $\dfrac{d }{d x} (f(x) \cdot g(x)) =f \cdot \dfrac{d g}{dx}+g \cdot \dfrac{d f}{dx}$ $\dfrac{d }{d x}(\dfrac{f(x)}{g(x)}) =\dfrac{g \cdot \dfrac{d f}{dx}-f \cdot \dfrac{d g}{d x}} {g^2}$ Chain Rule If $y=(f \circ g)(x)=f(g(x))$ and $u=g(x)$, then $\dfrac{d y}{d x}=\dfrac{d y}{d u} \dfrac{d u}{d x}$ Example 1: If $h(x)=(2x+1)^{10}$, what is $\dfrac {d h}{d x}$? Let $u(x)=2x+1$, hence $h(u)=u^{10}$ $\dfrac{d h}{d u}=10 u^9$, $\dfrac{d u}{d x}=2$ $\dfrac {d h}{d x}=\dfrac{d h}{d u} \dfrac{d u}{d x} = 10 u^9 \cdot 2 = 20(2x+1)^9$ Example 2: If $h(x)=\sin (x^2+5)$, what is $\dfrac {d h}{d x}$? Let $u(x)=x^2+5$, hence $h(u)=\sin (u)$ $\dfrac{d h}{d u}=\cos (u)$, $\dfrac{d u}{d x}=2 x$ $\dfrac {d h}{d x}=\dfrac{d h}{d u} \dfrac{d u}{d x} = \cos (u) \cdot 2 x = 2x \cos (x^2+5)$