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Chapter 3: Problem 45

Evaluate the derivative of the following functions at the given point. $$c=2 \sqrt{s}-1 ; s=25$$

Short Answer

Expert verified
Answer: The derivative of the function $$c(s) = 2\sqrt{s} - 1$$ at $$s=25$$ is $$\frac{1}{5}$$.

Step by step solution

01

Find the derivative of the function

To find the derivative of the function, we should find $$\frac{dc}{ds} = \frac{d( 2\sqrt{s}-1)}{ds}$$. Using the chain rule, the derivative of $$\sqrt{s}$$ with respect to $$s$$ is $$\frac{1}{2\sqrt{s}}$$. So, $$\frac{dc}{ds} = 2\frac{d\sqrt{s}}{ds} - 0$$ $$\frac{dc}{ds} = 2\left(\frac{1}{2\sqrt{s}}\right)$$
02

Simplify the derivative

Now, let's simplify the derivative we got in step 1: $$\frac{dc}{ds} = \frac{2}{2\sqrt{s}} = \frac{1}{\sqrt{s}}$$
03

Evaluate the derivative at $$s=25$$

Now, we can plug in the given value for $$s$$ into the derivative we found: $$\frac{dc}{ds}\Big|_{s=25} = \frac{1}{\sqrt{25}} = \frac{1}{5}$$ Therefore, the derivative of the function $$c(s) = 2\sqrt{s} - 1$$ at $$s=25$$ is $$\frac{1}{5}$$.

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