Question

Differentiate the function, and fin f(x)=4x+(3)/(x),x=-5

Answer 1

The derivative of the function
`\( f(x) = 4x + (3)/(x) \)` is
`\( f'(x) = 4 - (3)/(x^2) \),` and when evaluated at
`\( x = -5 \)` ,
`\( f'(-5) = 4 - (3)/((-5)^2) = (13)/(25) \).`

Step-by-step explanation:

To differentiate the function
`\( f(x) = 4x + (3)/(x) \),` we apply the rules of differentiation. The derivative of
`\(4x\) is \(4\),` and the derivative of
`\((3)/(x)\)` is
`\(-(3)/(x^2)\)` . Combining these results, we obtain
`\(f'(x) = 4 - (3)/(x^2)\).`

To find
`\(f'(-5)\),` we substitute
`\(x = -5\)` into the derivative expression, resulting in
`\(f'(-5) = 4 - (3)/((-5)^2) = 4 - (3)/(25) = (97)/(25)\).` Therefore, the derivative of
`\(f(x)\) at \(x = -5\) is \((13)/(25)\).`

Understanding the process of differentiation is fundamental in calculus. It allows us to find the rate at which a function is changing at a specific point. In this case, the derivative
`\(f'(x)\)` represents the slope of the tangent line to the graph of
`\(f(x)\)` , and evaluating it at
`\(x = -5\)` provides the instantaneous rate of change at that point.