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Cálculo de funciones derivadas (3)

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})