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Differentiating a function allows us to find out how the function is changing at any point. The first derivative of a function gives us the rate at which the function's value is changing concerning its input value. If we consider the function \( p(u) = -2.1u^3 + 3.5u^2 + 16 \), the first derivative \( p'(u) \) represents how \( p(u) \) changes as \( u \) changes.

To find the first derivative, we apply a rule known as the Power Rule of Differentiation. This rule is super important because it helps in simplifying the process of differentiation.

• The power rule states that for any term \( au^n \), its derivative is \( nau^{n-1} \), where \( a \) is the coefficient and \( n \) is the exponent.
• In our function \( -2.1u^3 \), applying the power rule: \(-2.1 \times 3 \times u^{3-1} = -6.3u^2\).
• For the \( 3.5u^2 \) term, the derivative becomes \( 3.5 \times 2 \times u^{2-1} = 7u \).
• The constant \( 16 \) disappears upon differentiation since its rate of change is zero.

Hence, the first derivative of the function \( p(u) = -2.1u^3 + 3.5u^2 + 16 \) is \( p'(u) = -6.3u^2 + 7u \).

Once we find the first derivative, we can take it a step further and find the second derivative. This gives us even more insight, particularly about the concavity of the original function and the acceleration of its change.

Once again, we use the power rule. Now, we differentiate the first derivative \( p'(u) = -6.3u^2 + 7u \).

• Apply the power rule to \( -6.3u^2 \): \( -6.3 \times 2 \times u^{2-1} = -12.6u \).
• For the \( 7u \) term, since it is a linear term \( 7u^1 \), its differentiation results in a constant \( 7 \).

So, the second derivative \( p''(u) \) of our original function is \( -12.6u + 7 \). This tells us about the acceleration of \( p(u) \)'s rate of change at any point \( u \). The second derivative can also reveal if the function is concave up or down at particular points.

The Power Rule of Differentiation is a fundamental tool in calculus that simplifies the process of finding derivatives. It states that for any term in the form of \( au^n \), the derivative is \( nau^{n-1} \). This rule is extremely useful when dealing with polynomial functions, like the one in our exercise.

Polynomials, such as \( -2.1u^3 + 3.5u^2 + 16 \), are made up of terms where variables are raised to a power and multiplied by a coefficient. The power rule allows you to take each term individually and apply the rule to find the derivative.

• Identify the exponent \( n \) and coefficient \( a \).
• Multiply \( a \) by \( n \).
• Lower the exponent \( n \) by one to get \( n-1 \).
• Write down the new term using the calculated coefficient and the new exponent.

Using the power rule makes deriving each term manageable and provides a quick way to find how a function behaves and changes over various values of its variable.