Question

3.3c differentiation rule
(6 points)
if ( g(t)=-2 t^{4}+4 t^{2}+9 ) find
( g(0)=)
( g^{prime}(0)=)
( g^{prime prime}(0)=)
( g^{prime prime prime}(0)=)
( g^{(4)}(0)=)
( g^{(5)}(0)=)
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Explanation:

Step1: Calculate \(g(0)\)

Substitute \(t = 0\) into \(g(t)=-2t^{4}+4t^{2}+9\).
\(g(0)=-2\times0^{4}+4\times0^{2}+9 = 9\)

Step2: Find the first - derivative \(g^{\prime}(t)\)

Using the power rule \((x^{n})^\prime=nx^{n - 1}\), \(g^{\prime}(t)=(-2t^{4}+4t^{2}+9)^\prime=-2\times4t^{3}+4\times2t+0=-8t^{3}+8t\)
Then substitute \(t = 0\) into \(g^{\prime}(t)\), \(g^{\prime}(0)=-8\times0^{3}+8\times0 = 0\)

Step3: Find the second - derivative \(g^{\prime\prime}(t)\)

Differentiate \(g^{\prime}(t)=-8t^{3}+8t\) using the power rule. \(g^{\prime\prime}(t)=(-8t^{3}+8t)^\prime=-8\times3t^{2}+8=-24t^{2}+8\)
Substitute \(t = 0\) into \(g^{\prime\prime}(t)\), \(g^{\prime\prime}(0)=-24\times0^{2}+8 = 8\)

Step4: Find the third - derivative \(g^{\prime\prime\prime}(t)\)

Differentiate \(g^{\prime\prime}(t)=-24t^{2}+8\) using the power rule. \(g^{\prime\prime\prime}(t)=(-24t^{2}+8)^\prime=-24\times2t=-48t\)
Substitute \(t = 0\) into \(g^{\prime\prime\prime}(t)\), \(g^{\prime\prime\prime}(0)=-48\times0 = 0\)

Step5: Find the fourth - derivative \(g^{(4)}(t)\)

Differentiate \(g^{\prime\prime\prime}(t)=-48t\) using the power rule. \(g^{(4)}(t)=(-48t)^\prime=-48\)
Since \(g^{(4)}(t)\) is a constant, \(g^{(4)}(0)=-48\)

Step6: Find the fifth - derivative \(g^{(5)}(t)\)

Differentiate \(g^{(4)}(t)=-48\) (a constant). The derivative of a constant is \(0\), so \(g^{(5)}(t)=0\) and \(g^{(5)}(0)=0\)

Answer:

\(g(0)=9\), \(g^{\prime}(0)=0\), \(g^{\prime\prime}(0)=8\), \(g^{\prime\prime\prime}(0)=0\), \(g^{(4)}(0)=-48\), \(g^{(5)}(0)=0\)