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Chapter 2: Problem 57

Calc Determine the following derivatives with respect to time \((t)\) : (a) \(\frac{d}{d t}\left(5 t^{2}+4 t+3\right)\) (b) \(\frac{d}{d t}\left(t^{2}-4 t-8\right)\)

Short Answer

Expert verified
The result of the derivatives are (a) \(10t + 4\) and (b) \(2t - 4\).

Step by step solution

01

- Apply Power and Constant Rule to Each Term of Equation (a)

Firstly, differentiate the function \(5t^2 + 4t+ 3\) with respect to \(t\). \n Using the power rule, the derivative of \(t^n\) is \(n \cdot t^{(n-1)}\). Also, the derivative of a constant is zero and the derivative of a term consisting a constant times a function is the constant times the derivative of that function. 'Thus the derivative of \(5t^2\) is \(2 \times 5 \times t^{2-1} = 10t\), the derivative of \(4t\) is \(4\), and the derivative of 3 is 0.
02

- Apply Power and Constant Rule to Each Term of Equation (b)

Now, differentiate the function \(t^2 - 4t- 8\) with respect to \(t\). Using similar rules as in step 1, the derivative of \(t^2\) is \(2 \times t^{2-1} = 2t\), the derivative of \(-4t\) is \(-4\), and the derivative of -8 is 0.
03

- Combine the Derivatives to Form the Final Result

Finally, combine the derivatives found in the previous steps and write in the original order of terms. The derivative of \(5t^2 + 4t+ 3\) is \(10t + 4\), and the derivative of \(t^2 - 4t - 8\) is \(2t - 4\).

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