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

  1. $\dfrac{d k}{d x}=0 $ , for any constant $k$

  2. $\dfrac{d }{d x}(k f)=k \dfrac{d f}{d x} $

  3. $\dfrac{d }{d x} x^n = n x^{n-1}$

  4. $\dfrac{d }{d x}[f(x)+g(x)] =\dfrac{d }{dx}f(x)+\dfrac{d}{dx}g(x)$

  5. $\dfrac{d }{d x} (f(x) \cdot g(x)) =f \cdot \dfrac{d g}{dx}+g \cdot \dfrac{d f}{dx}$

  6. $\dfrac{d }{d x}(\dfrac{f(x)}{g(x)}) =\dfrac{g \cdot \dfrac{d f}{dx}-f \cdot \dfrac{d g}{d x}} {g^2}$

Chain Rule



If $y=(f \circ g)(x)=f(g(x))$ and $u=g(x)$, then
  1. $\dfrac{d y}{d x}=\dfrac{d y}{d u} \dfrac{d u}{d x}$


    2. 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 $h(u)=\sin (u)$

    $\dfrac{d h}{d u}=\cos (u)$, $\dfrac{d u}{d x}=2 x$

    $\dfrac {d h}{d x}=\dfrac{d h}{d u} \dfrac{d u}{d x} = \cos (u) \cdot 2 x = 2x \cos (x^2+5)$
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