Calcula la función derivada de:
(a) $f(x)=3x^{4}-2x^{2}+5$ (d) $f(x)=\sqrt{x^{3}}+\sqrt[3]{x^{2}}$
(b) $f(x)=1-\dfrac{2}{x^{2}}$ (e) $f(x)=4(x^{2}+9x-13)$
(c) $f(x)=\sqrt{5x}$ (f) $f(x)=(3+7x+x^{3})^{2}$
Resolución:
(a) $f(x)=3x^{4}-2x^{2}+5$ $f^{'}(x)=12x^{3}-4x$
(b) $f(x)=1-\dfrac{2}{x^{2}}=1-2x^{-2}$ $f^{'}(x)=4x^{-3}=\dfrac{4}{x^{3}}$
(c) $f(x)=\sqrt{5x}=5^{\frac{1}{2}}x^{\frac{1}{2}}$ $f^{'}(x)=5^{\frac{1}{2}}\dfrac{1}{2}x^{\frac{1}{2}-1}=\dfrac{1}{2}5^{\frac{1}{2}}x^{-\frac{1}{2}}=\dfrac{\sqrt{5}}{2\sqrt{x}}$
(d) $f(x)=\sqrt{x^{3}}+\sqrt[3]{x^{2}}=x^{\frac{3}{2}}+x^{\frac{2}{3}}$
$f^{'}(x)=\dfrac{3}{2}x^{\frac{3}{2}-\frac{2}{2}}+\dfrac{2}{3}x^{\frac{2}{3}-\frac{3}{3}}=\dfrac{3}{2}x^{\frac{1}{2}}+\dfrac{2}{3}x^{-\frac{1}{3}}=\dfrac{3}{2}\sqrt{x}+\dfrac{2}{3\sqrt[3]{x}}$
(e) $f(x)=4(x^{2}+9x-13)$ $f^{'}(x)=4(2x+9)=8x+36$
(f) $f(x)=(3+7x+x^{3})^{2}$ $f^{'}(x)=2(3+7x+x^{3})(7+3x^{2})$
