In what follows $f$, $g$, and $h$ are functions of the variable x. k is a constant.
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$\displaystyle \dfrac{dy}{dx}=\lim_{h\to 0} \dfrac{f(x+h)-f(x)}{h}$ Other notations of the derivative are: $\dfrac{df}{dx}$ , $y'$ or $f'$.
Let $u(x)=2x+1$, hence $h(u)=u^{10}$ $\dfrac{d h}{d u}=10 u^9$, $\dfrac{d u}{d x}=2$ $\dfrac {d h}{d x}=\dfrac{d h}{d u} \dfrac{d u}{d x} = 10 u^9 \cdot 2 = 20(2x+1)^9$ Let $u(x)=x^2+5$, hence $h(u)=\sin (u)$ $\dfrac{d h}{d u}=\cos (u)$, $\dfrac{d u}{d x}=2 x$ $\dfrac {d h}{d x}=\dfrac{d h}{d u} \dfrac{d u}{d x} = \cos (u) \cdot 2 x = 2x \cos (x^2+5)$ |