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: bring the exponent down and reduce it by 1. Constants like stay multiplied out front.
2

Label answer for part a

This is the unsimplified derivative for part a. We leave as is rather than rewriting as a fraction, per the instructions.
3

Differentiate part b using Quotient Rule

The quotient rule is . Here so , and so . Plug straight in without simplifying.
4

Differentiate part c using Product Rule

The product rule is . Here and . We leave the result as two un-expanded terms.
5

Differentiate part d using Quotient Rule

Apply the quotient rule with , , , . Write the result directly from the formula without simplifying.
6

Differentiate parts e and f using trig rules

Standard trig derivatives: , , . Scalar multiples just stay out front. Leave unsimplified.
7

Differentiate part g using Product Rule

Product rule with , , , . The derivative of is . Result is left unsimplified.

Understanding this problem

Learning Insight

Every derivative rule — power, product, quotient — comes from the limit definition of the derivative. The product and quotient rules exist because two functions changing at the same time interact with each other, not independently. Recognizing WHICH rule to use (power vs. product vs. quotient) before computing is the most important skill here.

Quick Tip

For the quotient rule, remember 'low d-high minus high d-low, over low squared' — it helps you keep the subtraction in the right order every time.

Common Mistake

Students often forget the minus sign in the quotient rule numerator and write instead of , which flips the sign of the entire second term and gives a wrong answer.