Exercise
Find the derivative of the following function:
f ( x ) = 3 x 3 − 4 x 5 f(x)=\frac{3}{\sqrt[3]{x}}-\frac{4}{\sqrt{x^5}} f ( x ) = 3 x 3 − x 5 4
Final Answer
Show final answer
f'(x)=-\frac{1}{\sqrt[3]{x^4}}+10\cdot\frac{1}{\sqrt{x^7}}
Solution
We simplify the function before differentiating:
f ( x ) = 3 x 3 − 4 x 5 = f(x)=\frac{3}{\sqrt[3]{x}}-\frac{4}{\sqrt{x^5}}= f ( x ) = 3 x 3 − x 5 4 =
= 3 x − 1 3 − 4 x − 5 2 =3x^{-\frac{1}{3}}-4x^{-\frac{5}{2}} = 3 x − 3 1 − 4 x − 2 5
Using Derivative formulas
f ′ ( x ) = 3 ⋅ ( − 1 3 ) x − 4 3 − 4 ⋅ ( − 5 2 ) x − 7 2 f'(x)=3\cdot (-\frac{1}{3})x^{-\frac{4}{3}}-4\cdot (-\frac{5}{2})x^{-\frac{7}{2}} f ′ ( x ) = 3 ⋅ ( − 3 1 ) x − 3 4 − 4 ⋅ ( − 2 5 ) x − 2 7
One can simplify the derivative:
= − x − 4 3 + 10 x − 7 2 = =-x^{-\frac{4}{3}}+10x^{-\frac{7}{2}}= = − x − 3 4 + 1 0 x − 2 7 =
= − 1 x 4 3 + 10 ⋅ 1 x 7 =-\frac{1}{\sqrt[3]{x^4}}+10\cdot\frac{1}{\sqrt{x^7}} = − 3 x 4 1 + 1 0 ⋅ x 7 1
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