Calculus

Differentiate (find the derivative of) the function. Be sure to label correctly. DO NOT SIMPLIFY ANSWERS. (7 points each) a) f(x) = sqrt(x) - 1/((x^2)^(1/3)) (simplify first on the second term - do not use quotient rule) b) f(x) = x*sqrt(x) (simplify first - do not use product rule) c) f(x) = (x^3 - x^2 + 4)/x (simplify first - do not use quotient rule) d) f(x) = (7x^2 + x + 1)(3x + 2) (Use product rule) e) f(x) = (e^x + 8x)(4x^(-8) - x) (use product rule)

Step-by-step solution with explanation

Final Answer

a) | b) | c) | d) | e)

Step-by-step solution

1

Part a: Rewrite using exponent rules

We rewrite sqrt(x) as . For the second term, , so . Now we can use the power rule directly.
2

Part a: Apply power rule to each term

The power rule says: bring the exponent down front, then subtract 1 from the exponent. We apply this to each term separately. Do not simplify further.
3

Part b: Rewrite using exponent rules

We rewrite sqrt(x) as , then multiply by . When multiplying same bases, we add exponents: .
4

Part b: Apply power rule

Bring the exponent down front and subtract 1 from the exponent. Do not simplify further.
5

Part c: Rewrite by dividing each term by x

Split the fraction into three separate terms, each divided by . Simplify each using exponent rules: , , and .
6

Part c: Apply power rule to each term

Apply the power rule to each term. The derivative of is , the derivative of is , and the derivative of is . Do not simplify further.
7

Part d: Apply product rule to $(7x^2+x+1)(3x+2)$

The product rule states: . Here so , and so . Plug into the formula. Do not expand or simplify.
8

Part e: Apply product rule to $(e^x+8x)(4x^{-8}-x)$

Here so , and so . Apply the product rule: . Do not simplify further.

Understanding this problem

Learning Insight

Rewriting functions using exponent rules before differentiating is a powerful strategy. It lets you use the simple power rule — bring the exponent down, subtract 1 — instead of harder rules like the quotient or product rule. The product rule is needed when two separate functions of x are multiplied and cannot be combined into one term first.

Quick Tip

Before differentiating, always ask: can I rewrite this as a single term (or sum of terms) using exponent rules? If yes, just use the power rule and skip the quotient or product rule entirely.

Common Mistake

A very common mistake on part (a) is forgetting that , not or . Students often forget to multiply the exponents when raising a power to a power, which leads to the wrong base exponent before differentiating.