Question

3. Calcule, pela regra da cadeia, a derivada de y = (5x3-x4)7:

Answer 1

$y=f(x)=(5x^3-x^4)^7$


É uma função composta. Podemos enxergar assim:

$\left\{ \!\begin{array}{l} f(x)=\big(g(x)\big)^7\\\\ g(x)=5x^3-x^4 \end{array} \right.$

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Derivando $f$ usando a Regra da Cadeia:

$f'(x)=\Big[\big(g(x)\big)^7\Big]'\\\\\\ f'(x)=7\,\big(g(x)\big)^{7-1}\cdot g'(x)\\\\ f'(x)=7\,\big(g(x)\big)^{6}\cdot g'(x)\\\\ f'(x)=7\,(5x^3-x^4)^{6}\cdot (5x^3-x^4)'\\\\ f'(x)=7\,(5x^3-x^4)^{6}\cdot (5\cdot 3x^{3-1}-4x^{4-1})\\\\ \boxed{\begin{array}{c}f'(x)=7\,(5x^3-x^4)^{6}\cdot (15x^2-4x^3) \end{array}}$