Question

Simplify the following expression: d/dx∫dp/p²

Answer 1

The simplified expression of
`\( (d)/(dx)\int (dp)/(p^2) \) is \( -2(d)/(dx) \left((1)/(p)\right) \).`The initial integral is evaluated, yielding
`\( -(1)/(p) \),`and then differentiating this result with respect to \( x \) gives the final simplified expression,
`\( -2(d)/(dx)\left((1)/(p)\right) \).`

Explanation:

The expression
`\( (d)/(dx)\int (dp)/(p^2) \)`involves differentiating the integral of
`\( (1)/(p^2) \)`with respect to \( x \). According to the fundamental theorem of calculus, differentiating an integral with respect to x can be approached by differentiating the antiderivative with respect to x . The integral
`\( \int (dp)/(p^2) \) yields \( -(1)/(p) \), and the derivative of \( -(1)/(p) \)`with respect to x is
`\( -2(d)/(dx)\left((1)/(p)\right) \)`. This result simplifies the original expression.

This simplification is derived by applying the fundamental theorem of calculus, involving the differentiation of an integral. The initial integral is evaluated, yielding
`\( -(1)/(p) \),`and then differentiating this result with respect to \( x \) gives the final simplified expression,
`\( -2(d)/(dx)\left((1)/(p)\right) \).`