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

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

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

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

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derivative of

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

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

Solution steps

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

Apply the Product Rule:

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

f = (t + 1)^{\frac{2}{3}}, g = (3 t^{2} - 2)^{3}

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

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{e ^{x}}{1 + e ^{x}})

3 e^{5 x} \frac{d y}{d x} = - 25 \frac{x}{y ^{2}}

\int 2 tan (θ) d θ

\int 4 sin^{3} (x co) s^{5} x d x

2 e^{2}

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