Derivative rules: flashcards, quiz and summary

Flashcards

1. 1What does the derivative f′(x) tell you about the function f at a point x?

   It gives the instantaneous rate of change of f at x. On a graph, it is the slope of the tangent line at that point.

2. 2How do you write the tangent line to f at x = a?

   Use y − f(a) = f′(a)(x − a). The slope is f′(a) and the line passes through the point (a, f(a)).

3. 3What is the power rule, and does it work for negative or fractional exponents?

   d/dx[xⁿ] = n·xⁿ⁻¹. Yes, it works for negative and fractional exponents too, so d/dx[√x] = 1/(2√x).

4. 4When do you use the product rule, and why is the derivative of a product not just the product of the derivatives?

   Use it when two functions are multiplied: d/dx[f·g] = f′g + fg′. The derivative of a product is not the product of the derivatives because each function's change affects the whole product.

5. 5What is the quotient rule, and why does the order of its numerator matter?

   d/dx[f/g] = (f′g − fg′) / g². The order matters because subtraction is not commutative, so swapping the terms flips the sign of the answer.

6. 6How does the chain rule work for a function inside another function?

   d/dx[f(g(x))] = f′(g(x))·g′(x). Differentiate the outside function, leave the inside unchanged, then multiply by the derivative of the inside.

7. 7What are the derivatives of sin x and cos x?

   d/dx[sin x] = cos x and d/dx[cos x] = −sin x. The sign on cos x's derivative is negative.

8. 8What is the derivative of eˣ, and why can't you use the power rule on it?

   d/dx[eˣ] = eˣ. The power rule does not apply because x is in the exponent, not the base, so eˣ is not a power function of x.

9. 9What does the sign of f′(x) tell you about the graph of f?

   If f′(x) > 0, then f is increasing. If f′(x) < 0, then f is decreasing.

10. 10What are critical points, and why do they matter when looking for maximums and minimums?

    Critical points are values of x where f′(x) = 0 or is undefined. They are candidates for maximums and minimums, so you check them with the sign of f′ or with f″.

11. 11What does the sign of the second derivative f″(x) tell you about a curve?

    If f″(x) > 0, the curve is concave up. If f″(x) < 0, the curve is concave down.

Practice quiz

Pick an answer to check it.

1. 1What is d/dx[x³·eˣ]?

   1. 3x²·eˣ
   2. 3x²·eˣ − x³·eˣ
   3. 3x²·eˣ + x³·eˣ
   4. x³·eˣ + 3x²
   Show the answer

   C. 3x²·eˣ + x³·eˣ This is a product, so use the product rule: f′g + fg′ with f = x³ and g = eˣ. That gives 3x²·eˣ + x³·eˣ. The choice 3x²·eˣ multiplies the derivatives, which is not allowed.

2. 2What is d/dx[2x / (x − 1)]?

   1. −2/(x − 1)²
   2. 2/(x − 1)²
   3. −2x/(x − 1)²
   4. 2(x − 1)/(x − 1)²
   Show the answer

   A. −2/(x − 1)² Use the quotient rule: [2(x − 1) − 2x·1] / (x − 1)². The numerator simplifies to 2x − 2 − 2x = −2, so the answer is −2/(x − 1)².

3. 3What is d/dx[(x² + 5)³]?

   1. 3(x² + 5)²
   2. 3x²(x² + 5)²
   3. 6x(x² + 5)³
   4. 6x(x² + 5)²
   Show the answer

   D. 6x(x² + 5)² This is a chain rule problem. Differentiate the outside to get 3(x² + 5)², then multiply by the derivative of the inside, 2x, to get 6x(x² + 5)². Leaving out the inside derivative is the most common mistake.

4. 4What is d/dx[sin x − cos x]?

   1. cos x + sin x
   2. sin x − cos x
   3. cos x − sin x
   4. −cos x − sin x
   Show the answer

   A. cos x + sin x The derivative of sin x is cos x, and the derivative of −cos x is −(−sin x) = sin x. Adding them gives cos x + sin x. The sign change on cos x is the trap.

5. 5What is d/dx[4√x]?

   1. 1/(2√x)
   2. 4/√x
   3. 2/√x
   4. 2√x
   Show the answer

   C. 2/√x Write 4√x as 4x^(1/2). The power rule gives 4·(1/2)x^(−1/2) = 2x^(−1/2) = 2/√x. Forgetting the constant multiple or misapplying the exponent gives the other choices.

6. 6Where are the critical points of f(x) = x³ − 3x?

   1. x = 1 only
   2. x = 0 only
   3. x = 3 and x = −3
   4. x = 1 and x = −1
   Show the answer

   D. x = 1 and x = −1 f′(x) = 3x² − 3. Setting it equal to 0 gives x² = 1, so x = 1 or x = −1. Both are critical points because f′ equals 0 there.

7. 7What is d/dx[e^(2x)]?

   1. e^(2x)/2
   2. 2e^(2x)
   3. 2x·e^(2x−1)
   4. e^(2x)
   Show the answer

   B. 2e^(2x) This is the chain rule with the exponential function. The derivative of eˣ is eˣ, so the outside stays e^(2x), and the inside derivative is 2. The product is 2e^(2x).

Test traps

What teachers like to test on this, and the mistakes students make.

• Forgetting the chain rule's inside derivative, so d/dx[(x² + 5)³] is written as 3(x² + 5)² instead of 6x(x² + 5)².
• Reversing the quotient rule numerator, so f′g − fg′ becomes fg′ − f′g and the sign of the answer flips.
• Multiplying the derivatives in a product, writing d/dx[f·g] = f′·g′, instead of using f′g + fg′.
• Dropping the negative sign in d/dx[cos x] = −sin x, which is one of the most common lost points on tests.
• Applying the power rule to eˣ or to aˣ, when the exponent is the variable and the rule does not apply.
• Confusing a zero of f with a zero of f′: critical points come from f′(x) = 0, not f(x) = 0.

Summary

• The derivative f′(x) gives the instantaneous rate of change of f at x and equals the slope of the tangent line at that point.
• The power rule says d/dx[xⁿ] = n·xⁿ⁻¹, and it also works for negative and fractional powers, such as d/dx[√x] = 1/(2√x).
• The product rule (f′g + fg′) and the quotient rule ((f′g − fg′)/g²) handle products and quotients; the derivative of a product is not the product of the derivatives.
• The chain rule handles a function inside a function: differentiate the outside, keep the inside unchanged, then multiply by the derivative of the inside.
• The derivatives of sin x and cos x are cos x and −sin x, and the derivative of tan x is sec² x; these only hold when angles are in radians.
• The derivative of eˣ is eˣ, and the derivative of ln x is 1/x, while aˣ differentiates to aˣ·ln a.
• The sign of f′(x) tells you whether f is increasing (positive) or decreasing (negative), and critical points occur where f′(x) = 0 or is undefined.
• The sign of f″(x) tells you concavity: positive means concave up and negative means concave down.

Key terms

Derivative f′(x)

The instantaneous rate of change of f at x, defined as the limit of [f(x + h) − f(x)] / h as h approaches 0.

Tangent line

The line that touches a curve at a point and has the same slope as the curve there. Its equation is y − f(a) = f′(a)(x − a).

Power rule

d/dx[xⁿ] = n·xⁿ⁻¹. It works for negative and fractional exponents as well as positive integers.

Product rule

d/dx[f·g] = f′g + fg′. Use it when two functions are multiplied together.

Quotient rule

d/dx[f/g] = (f′g − fg′) / g². The order in the numerator matters: f′g comes first, then fg′.

Chain rule

d/dx[f(g(x))] = f′(g(x))·g′(x). Use it when one function sits inside another.

Critical point

A value of x where f′(x) = 0 or f′(x) is undefined. These are candidates for maximums and minimums.

Concavity

The shape of a graph described by f″(x). Concave up when f″(x) > 0 and concave down when f″(x) < 0.

How to study it

The notes this pack came from

Questions