Differentiating Functions Using the Power Rule

1. \(\dfrac{d}{ds}(s^{3}-2s+1) ^{5/3}=\dfrac{5}{3} (s^{3}-2s+1) ^{2/3}\dfrac{d}{ds}(s^{3}-2s+1) = \dfrac{5}{3}(s^{3}-2s+1) ^{2/3}(3s^{2}-2) \)
2. \(\begin{align*} \dfrac{d}{dx}\sqrt[3]{x^{4}-3x+5} &=\dfrac{d}{dx}(x^{4}-3x+5) ^{1/3}=\dfrac{1}{3}(x^{4}-3x+5) ^{-2/3}\dfrac{d}{dx}(x^{4}-3x+5) \\&= \dfrac{4x^{3}-3}{3(x^{4}-3x+5)^{2/3}} \end{align*}\)
3. \(\begin{align*} \dfrac{d}{d\theta }[\tan (3\theta ) ] ^{-3/4}&=-\dfrac{3}{4}[\tan (3\theta ) ] ^{-7/4} \dfrac{d}{d\theta }\tan (3\theta ) = -\dfrac{3}{4}[\tan (3\theta ) ] ^{-7/4}\cdot \sec ^{2}(3\theta ) \cdot 3 \\&= -\dfrac{9\sec^2 (3\theta ) }{4[\tan (3\theta ) ] ^{7/4}} \end{align*}\)