To find the velocity function based on displacement, use the first derivative of f(x) = 2^1.25x+ x². f'(x) = 1.25 * ln(2) * 2^1.25x+ 2x

The velocity at x = 10 would therefore be: f'(10) = 1.25 * log(2) * 2^{1.25 * 10}+ 2 * 10 = 1.25 * 2^12.5* log(2) + 20 feet/second

The instantaneous velocity is represented by the first derivative of the positional equation.

v(t) = s'(t) = 12 * (1/2) sec²(t/2 + π) = 6sec²(t/2 + π) = 6/cos²(t/2 + π) = 6/((–1)²cos²(t/2)) = 6/cos²(t/2)

Based on the nature of the cosine, we know that 6/cos²(t/2 + π) = 6/((–1)²cos²(t/2)) = 6/cos²(t/2)

To solve, v(π/2) = 6/cos²(t/2) = 6/((1/√2)²) = 6/(1/2) = 12

The instantaneous velocity is represented by the first derivative of the positional equation. This is found by using the chain rule both on the square of the cosine function and the function itself.

v(t) = s'(t) = 12 * 2 cos(t/2 + π) * (–sin(t/2 + π)) * (1/2) = –12cos(t/2 + π)sin(t/2 + π)

Given what we know about the cosine and sine functions, we know cos(t/2 + π) = –cos(t/2) and sin(t/2 + π) = –sin(t/2)

Therefore, v(t) = –12(–sin(t/2))(–cos(t/2)) = –12sin(t/2)cos(t/2)

v(π/2) = –12sin(π/4)cos(π/4) = –12(1/√2)(1/√2) = –12 * (1/2) = –6

The velocity function is given by the derivative of the position function. So here. Plugging 3 in forgives 16.

The velocity is given as the derivative of the position function, or

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We can use the quotient rule to find the derivative of the position function and then evaluate that at. The quotient rule states that

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In this case,and.

We can now substitute these values in to get

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Evalusting this atgives us.

So the answer is.

有三个在这probl条款em that has to be derived. The derivative of the position function, or the velocity function, represents the slope of the position function.

The derivative ofcan be solved by using the power rule, which is:

Therefore. the derivative ofis.

The derivative ofisby using the constant multiple rule.

The derivative ofis由于衍生品of constants equal to zero.

The derivatives can be solved term by term.

First, find the derivative of. This can be done by the power rule.

Find the derivative of. Rewrite this asto compute by power rule.

Therefore, the velocity function is:

To find the velocity given the acceleration function, we will need to integrate the acceleration function.

The velocity function can be obtained by integrating the acceleration function.

Since we are finding the velocity at, substitute this into the velocity function.