Calculus

Suppose that f(2) = -1, g(2) = 5, f'(2) = -4, and g'(2) = 3, find the following: a) (f - g)'(2) b) (fg)'(2) c) (f/g)'(2) d) (f/(f + g))'(2)

Step-by-step solution with explanation

Final Answer

a) b) c) d)

Step-by-step solution

1

List all given values clearly

We write out every value we know. We'll plug these into derivative rules for each part.
2

Part a: Difference Rule for derivatives

The difference rule says . We just subtract the two derivative values at .
3

Part b: Product Rule for derivatives

The product rule says . We plug in all four known values and simplify.
4

Part c: Quotient Rule for f/g

\left(\frac{f}{g}\right)'(2) = \frac{f'(2)\cdot g(2) - f(2)\cdot g'(2)}{[g(2)]^2} = \frac{(-4)(5) - (-1)(3)}{5^2} = \frac{-20 + 3}{25} = \frac{-17}{25}$
The quotient rule says . We plug in all values and simplify the fraction.
5

Part d: Quotient Rule for f/(f+g)

The denominator here is , so we treat it as one function. We apply the quotient rule: top is , bottom is . Note .
6

Plug in values for part d

We compute each bracket step by step: , , then multiply and subtract. The bottom is .

Understanding this problem

Learning Insight

These four rules — sum/difference, product, and quotient — let you find derivatives of combined functions without knowing the full formula for or . You only need the values at the specific point. This is powerful because real-world functions are often defined only by data points, not formulas.

Quick Tip

For the quotient rule, remember 'low d-high minus high d-low, over low squared' — that phrase keeps the order of subtraction correct every time.

Common Mistake

On the quotient rule, students often flip the subtraction and write instead of in the numerator. The derivative of the TOP goes first — getting this backward gives the wrong sign on your answer.