Differential relationships among load, shear, and bending moment Which statements correctly express the fundamental relations for a beam (with consistent sign conventions)?

Introduction / Context:
The shear-force and bending-moment diagrams arise directly from equilibrium applied locally to a beam element. Remembering the differential relations helps locate maximum M and relate changes in V to the applied load intensity.

Given Data / Assumptions:

• Transverse loading on a slender beam.
• Consistent sign convention (e.g., sagging M positive; downward w positive).

Concept / Approach:
The governing relationships are:
dM/dx = VdV/dx = wThese are obtained by taking moments and forces on a differential element and letting the element length tend to zero. They are independent of material properties and are purely from statics (with the chosen sign convention).

Step-by-Step Solution:

Verification / Alternative check:
Integrating w over a span gives the change in shear, and integrating V gives the change in bending moment — exactly how SFD and BMD are constructed.

Why Other Options Are Wrong:

• Only one correct (options a or b) ignores the other true relation.
• Neither correct (option d) contradicts statics.
• Option e swaps the relations; that is incorrect.

Common Pitfalls:
Sign mistakes when drawing diagrams; always stick to a consistent convention and apply the differential relations carefully.

Final Answer:
Both dM/dx = V and dV/dx = w are correct