Ejercicio 8. Sea \( f_{n}(x)=x^{n} \), para \( x \in \mathbb{R} \) y \( n \in \mathbb{N} \). Demuestra por inducción que \( f_{n}^{\prime}(x)=n x^{n-1} \).
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- The function is \( f_1(x) = x^1 = x \). - The derivative is \( f_1'(x) = 1 \). - The formula \( n x^{n-1} \) gives \( 1 \cdot x^{1-1} = 1 \). - The base case holds: \( f_1'(x) = 1 \). Show more…
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If $f(x)=x,$ then $f^{\prime}(x)=1 \cdot x^{0}=1$ If $f(x)=x^{2},$ then $f^{\prime}(x)=2 x^{\prime}-2 x$ (a) Show that $$f(x)=x^{3}, \quad \text { then } \quad f^{\prime}(x)=3 x^{2}$$ (b) Prove by induction that for each positive integer $n$ $$f(x)=x^{n} \quad \text { has derivative } \quad f^{\prime}(x)=n x^{n-1}$$
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Zhumagali S.
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