What is the d/(dt)((y^'(t))^2) ?

The d/(dt)((y^'(t))^2) is 2ty^{2'}+2'(dy)/(dt)y^{2'-1}t^2

What is the first d/(dt)((y^'(t))^2) ?

The first d/(dt)((y^'(t))^2) is 2ty^{2'}+2'(dy)/(dt)y^{2'-1}t^2

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\frac{d}{d t} ((y^{^{'}} (t))^{2})

2 t y^{2'} + 2' \frac{d y}{d t} y^{2' - 1} t^{2}

Solution steps

\frac{d}{d t} ((y^{^{'}} (t))^{2})

Simplify

(y^{^{'}} (t))^{2} : t^{2} y^{2^{'}}

= \frac{d}{d t} (t^{2} y^{2^{^{'}}})

Apply the Product Rule:

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

f = t^{2}, g = y^{2^{'}}

= \frac{d}{d t} (t^{2}) y^{2^{^{'}}} + \frac{d}{d t} (y^{2^{^{'}}}) t^{2}

\frac{d}{d t} (t^{2}) = 2 t

\frac{d}{d t} (y^{2^{'}}) = 2^{'} \frac{d y}{d t} y^{2^{'} - 1}

= 2 t y^{2^{^{'}}} + 2^{^{'}} \frac{d y}{d t} y^{2^{^{'}} - 1} t^{2}

(3 x) (6 x^{2})

- 4 a^{4} b^{4}

7 e^{- 93 \cdot 34}

5.3 x 1 0^{- 3} m^{3}

What is the d/(dt)((y^'(t))^2) ?
The d/(dt)((y^'(t))^2) is 2ty^{2'}+2'(dy)/(dt)y^{2'-1}t^2
What is the first d/(dt)((y^'(t))^2) ?
The first d/(dt)((y^'(t))^2) is 2ty^{2'}+2'(dy)/(dt)y^{2'-1}t^2