What is the derivative of (t+5)^{2/3}(3t^2-2)^3 ?

The derivative of (t+5)^{2/3}(3t^2-2)^3 is (2(3t^2-2)^3+54t(3t^2-2)^2(t+5))/(3(t+5)^{1/3)}

What is the first derivative of (t+5)^{2/3}(3t^2-2)^3 ?

The first derivative of (t+5)^{2/3}(3t^2-2)^3 is (2(3t^2-2)^3+54t(3t^2-2)^2(t+5))/(3(t+5)^{1/3)}

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derivative of

(t + 5)^{\frac{2}{3}} (3 t^{2} - 2)^{3}

\frac{2 ( 3 t ^{2} - 2 ) ^{3} + 54 t ( 3 t ^{2} - 2 ) ^{2} ( t + 5 )}{3 ( t + 5 ) ^{\frac{1}{3}}}

Solution steps

\frac{d}{d t} ((t + 5)^{\frac{2}{3}} (3 t^{2} - 2)^{3})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = (t + 5)^{\frac{2}{3}}, g = (3 t^{2} - 2)^{3}

= \frac{d}{d t} ((t + 5)^{\frac{2}{3}}) (3 t^{2} - 2)^{3} + \frac{d}{d t} ((3 t^{2} - 2)^{3}) (t + 5)^{\frac{2}{3}}

\frac{d}{d t} ((t + 5)^{\frac{2}{3}}) = \frac{2}{3 ( t + 5 ) ^{\frac{1}{3}}}

\frac{d}{d t} ((3 t^{2} - 2)^{3}) = 18 t (3 t^{2} - 2)^{2}

= \frac{2}{3 ( t + 5 ) ^{\frac{1}{3}}} (3 t^{2} - 2)^{3} + 18 t (3 t^{2} - 2)^{2} (t + 5)^{\frac{2}{3}}

Simplify

\frac{2}{3 ( t + 5 ) ^{\frac{1}{3}}} (3 t^{2} - 2)^{3} + 18 t (3 t^{2} - 2)^{2} (t + 5)^{\frac{2}{3}} : \frac{2 ( 3 t ^{2} - 2 ) ^{3} + 54 t ( 3 t ^{2} - 2 ) ^{2} ( t + 5 )}{3 ( t + 5 ) ^{\frac{1}{3}}}

= \frac{2 ( 3 t ^{2} - 2 ) ^{3} + 54 t ( 3 t ^{2} - 2 ) ^{2} ( t + 5 )}{3 ( t + 5 ) ^{\frac{1}{3}}}

\frac{d y}{d x} + 5 y = 70, y (1) = 110

x \to 10 lim (\frac{7 x ^{2} - 70 x}{x ^{2} - 100})

\int cos (0.5 x) d x

e^{- \frac{x}{2}}

t e^{t}

What is the derivative of (t+5)^{2/3}(3t^2-2)^3 ?
The derivative of (t+5)^{2/3}(3t^2-2)^3 is (2(3t^2-2)^3+54t(3t^2-2)^2(t+5))/(3(t+5)^{1/3)}
What is the first derivative of (t+5)^{2/3}(3t^2-2)^3 ?
The first derivative of (t+5)^{2/3}(3t^2-2)^3 is (2(3t^2-2)^3+54t(3t^2-2)^2(t+5))/(3(t+5)^{1/3)}