Question - Derivative of a Polynomial
Solution:
$$\frac{dy}{dt} = \frac{d}{dt}(2t^4) - \frac{d}{dt}(10t^2) + \frac{d}{dt}(13t) \\\frac{dy}{dt} = 8t^3 - 20t + 13$$
Tutor Solution:
To find the derivative of the given polynomial function $$ y = 2t^4 - 10t^2 + 13t $$, we will apply the power rule for derivatives. The power rule states that the derivative of $$ t^n $$ with respect to $$ t $$ is $$ nt^{n-1} $$. Let's find the derivative term by term:1. For the first term $$ 2t^4 $$, we apply the power rule: The derivative is $$ 4 \times 2t^{4-1} = 8t^3 $$.2. For the second term $$ -10t^2 $$, we again apply the power rule:The derivative is $$ 2 \times -10t^{2-1} = -20t $$.3. For the third term $$ 13t $$, the power rule gives us:The derivative is $$ 1 \times 13t^{1-1} = 13 $$.Now, combine all the derivatives of the individual terms to get the complete derivative of the function.The derivative of $$ y $$ with respect to $$ t $$ is: \[ \frac{dy}{dt} = 8t^3 - 20t + 13 \]This result represents the rate of change of $$ y $$ with respect to $$ t $$. Each term's coefficient in the original function gets multiplied by the exponent, and the new exponent is one less than the original. The constant term becomes 0 after differentiation since the derivative of a constant term with respect to $$ t $$ is always 0. However, since there is no constant term in our original function, we do not have to worry about that here. The final result simply sums the derivatives of all the terms.
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