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})