Question

If h(2) = 3 and h'(2) = -7, find d/dx(h(x)/x) x = 2.

Answer 1

Answer: -17/4

Work Shown

`\frac{d}{d\text{x}}\left(\frac{h(\text{x})}{\text{x}}\right) = \frac{\frac{d}{d\text{x}}(h(\text{x}))*\text{x}-h(\text{x})*\frac{d}{d\text{x}}(\text{x})}{\text{x}^2} \ \text{ .... quotient rule}\\\\\frac{d}{d\text{x}}\left(\frac{h(\text{x})}{\text{x}}\right) = \frac{h'(\text{x})*\text{x} - h(\text{x})}{\text{x}^2}\\\\`

Evaluate that at x = 2.

`\frac{h'(\text{x})*\text{x} - h(\text{x})}{\text{x}^2}\\\\=(h'(2)*2 - h(2))/(2^2)\\\\=(-7*2 - 3)/(2^2)\\\\=-(17)/(4)\\\\`

Therefore,

`\frac{d}{d\text{x}}\left(\frac{h(\text{x})}{\text{x}}\right)=-(17)/(4) \ \text{ when h(2) = 3, h'(2) = -7, and x = 2}\\\\`