Calculus

Find the derivative of the function. PLEASE do not simplify. Label. a) f(x) = (1/2)*pi*x^5 - 1/x^6 b) f(x) = x/(4 - x^3) c) f(x) = (4x^2 - 3x + 1)(-2x + 7) d) f(x) = cot(x)/sin(x) e) f(x) = -3cos(x) + 4sin(x) f) f(x) = tan(x) + cos(x) g) f(x) = x^6 * sec(x)

Step-by-step solution with explanation

Final Answer

a) | b) | c) | d) | e) | f) | g)

Step-by-step solution

1

Differentiate part a using power rule

f(x) = \frac{1}{2}\pi x^5 - x^{-6}f'(x) = \frac{1}{2}\pi \cdot 5x^4 - (-6)x^{-7}
Rewrite as so the power rule applies. The power rule says . Pi is just a constant multiplier, so it stays.
2

Write the unsimplified answer for part a

We leave this exactly as the power rule gives it — no combining or rewriting. This is the labeled, unsimplified derivative for part a.
3

Differentiate part b using the quotient rule

The quotient rule is . Here , , , . We plug in directly without simplifying.
4

Differentiate part c using the product rule

The product rule is . Here , , , . We write the result directly without expanding.
5

Differentiate part d using the quotient rule

Apply the quotient rule with , , , . Plug directly into without simplifying.
6

Differentiate parts e and f using trig rules

Basic trig derivatives: , , . Constants multiply through. Results are left unsimplified.
7

Differentiate part g using the product rule

Product rule with , , , . Write directly. Do not factor or simplify.

Understanding this problem

Learning Insight

The product and quotient rules both come from the same limit definition of the derivative — they are built-in patterns for when two functions are multiplied or divided. Knowing WHICH rule to reach for (power, product, quotient, trig) before you start is the real skill being practiced here.

Quick Tip

Quick rule selector: one fraction → quotient rule; two things multiplied → product rule; single term with exponent → power rule; lone trig function → memorized trig derivative. Label it BEFORE you differentiate.

Common Mistake

On the quotient rule, students frequently write in the numerator instead of , or they forget the minus sign between the two terms — always write the template first, then fill in.