QQuestionMathematics
QuestionMathematics
Derivative for the Volume of a Cylinder:
If I take the derivative with respect to r, is h treated as a constant?
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Answer
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Step 1:I'll solve this step-by-step, focusing on the derivative of the cylinder's volume with respect to its radius.
Step 2:: Recall the Volume Formula for a Cylinder
- $$h$$ is the height
The volume of a cylinder is given by: Where:
Step 3:: Differentiation with Respect to r
When differentiating $$V = \pi r^{2}h$$, and treating $$h$$ as a constant:
\frac{dV}{dr} = \pi \cdot 2r \cdot h
Step 4:: Explanation of Constant Treatment
- The derivative becomes $$2\pi rh
- This follows the basic rules of differentiation where non-varying terms are treated as constants
Final Answer
When differentiating the volume of a cylinder with respect to radius r, the height h is treated as a constant, resulting in the derivative \frac{dV}{dr} = 2\pi rh.
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