AP Calculus AB · Flashcards
Unit 2 — Differentiation: Definition & Rules
Course framework: Unit 2, plus the chain rule and implicit differentiation from Unit 3 · 58 cards. Answer each card in your head, flip it, then mark it Got it or Review again. Your progress is saved in this browser.
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Derivative Definition✓ Known #1
State the definition of the derivative at \(x=a\).
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Answer#1
\(f'(a)=\displaystyle\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\), if the limit exists.
Trap: Writing the formula without “if the limit exists.”
Derivative Definition✓ Known #2
What does \(f'(a)\) represent?
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Answer#2
- The slope of the tangent line to \(y=f(x)\) at \(x=a\).
- The instantaneous rate of change of \(f\) at \(a\).
Trap: Confusing the tangent slope with a secant slope.
Derivative Definition✓ Known #3
Alternative definition: write \(f'(a)\) using \(x\to a\).
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Answer#3
\(f'(a)=\displaystyle\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\).
Trap: Using \(a-x\) in the denominator without fixing the sign.
Derivative Definition✓ Known #4
What does \(\dfrac{f(a+h)-f(a)}{h}\) represent?
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Answer#4
The slope of the secant line through \((a,f(a))\) and \((a+h,f(a+h))\): an average rate of change.
Trap: Calling it a tangent slope before taking the limit.
Derivative Definition✓ Known #5
If \(f'(a)\) exists, what must be true about \(f\) at \(a\)?
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Answer#5
\(f\) is continuous at \(a\): differentiability implies continuity.
Trap: Continuity does not guarantee differentiability.
Derivative Definition✓ Known #6
Three graph features where a continuous function is not differentiable:
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Answer#6
- Corner
- Cusp
- Vertical tangent
Trap: Forgetting that any discontinuity also rules out differentiability.
Derivative Definition✓ Known #7
To show differentiability at \(a\) with limits, what must you verify?
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Answer#7
- Compute \(\displaystyle\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\).
- Confirm the limit exists: finite and equal from both sides.
Trap: Stopping after simplifying without checking that the limit exists.
Derivative Definition✓ Known #8
If the left- and right-hand difference-quotient limits differ at \(a\), what does that mean?
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Answer#8
\(f\) is not differentiable at \(a\) — typically a corner.
Trap: Saying “the derivative is 0” or averaging the two slopes.
Derivative Definition✓ Known #9
If \(f'(a)=0\), what does that say about the tangent line at \(x=a\)?
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Answer#9
The tangent line is horizontal (slope 0).
Trap: Assuming a horizontal tangent always means a maximum or minimum.
Derivative Definition✓ Known #10
Differentiate from first principles: \(f(x)=x^2\).
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Answer#10
- \(f'(x)=\displaystyle\lim_{h\to0}\frac{(x+h)^2-x^2}{h}\)
- \(=\displaystyle\lim_{h\to0}\frac{2xh+h^2}{h}=\lim_{h\to0}(2x+h)=2x\)
Trap: Expanding \((x+h)^2\) as \(x^2+h^2\).
Derivative Definition✓ Known #11
Which derivative does \(\displaystyle\lim_{h\to0}\frac{\sin\left(\frac{\pi}{2}+h\right)-1}{h}\) represent, and what is its value?
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Answer#11
- It is \(f’\left(\frac{\pi}{2}\right)\) for \(f(x)=\sin x\).
- Value: \(\cos\frac{\pi}{2}=0\).
Trap: Trying to evaluate the limit directly instead of recognizing the definition.
Derivative Definition✓ Known #12
Write the equation of the tangent line to \(y=f(x)\) at \(x=a\).
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Answer#12
\(y-f(a)=f'(a)(x-a)\), or \(y=f(a)+f'(a)(x-a)\).
Trap: Using \(f'(x)\) instead of the number \(f'(a)\) as the slope.
Derivative Definition✓ Known #13
A table gives \(f(1.9)=3.8\) and \(f(2.1)=4.6\). Estimate \(f'(2)\).
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Answer#13
Use the symmetric difference quotient: \(f'(2)\approx\dfrac{4.6-3.8}{2.1-1.9}=\dfrac{0.8}{0.2}=4\).
Trap: Dividing by \(0.1\) instead of the full interval \(0.2\).
Basic Rules✓ Known #14
Power rule: if \(f(x)=x^n\), what is \(f'(x)\)?
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Answer#14
\(f'(x)=n x^{n-1}\) (for any real exponent \(n\)).
Trap: Writing \(nx^n\).
Basic Rules✓ Known #15
Constant rule: if \(f(x)=c\), what is \(f'(x)\)?
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Answer#15
\(0\).
Trap: Writing \(c\).
Basic Rules✓ Known #16
Constant multiple rule: if \(f(x)=c\cdot g(x)\), what is \(f'(x)\)?
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Answer#16
\(c\cdot g'(x)\).
Trap: Differentiating the constant as well.
Basic Rules✓ Known #17
Sum/difference rule: what is the derivative of \(f(x)\pm g(x)\)?
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Answer#17
\((f\pm g)’=f’\pm g’\).
Trap: Dropping the minus sign.
Basic Rules✓ Known #18
Differentiate \(x^{-3}\).
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Answer#18
\(-3x^{-4}\).
Trap: Lowering the exponent to \(-2\) instead of \(-4\).
Basic Rules✓ Known #19
Differentiate \(\sqrt{x}\).
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Answer#19
\(\dfrac{1}{2\sqrt{x}}\), from \(x^{1/2}\to\frac12x^{-1/2}\).
Trap: Missing the \(\frac12\).
Basic Rules✓ Known #20
Differentiate \(\dfrac{1}{x^5}\).
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Answer#20
- Rewrite: \(x^{-5}\).
- Derivative: \(-5x^{-6}=-\dfrac{5}{x^6}\).
Trap: Treating it like \(\frac{1}{5x^4}\).
Basic Rules✓ Known #21
What is \(\dfrac{d}{dx}(x)\)?
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Answer#21
\(1\).
Trap: Writing \(0\).
Basic Rules✓ Known #22
If \(f(x)=3x^4-7x+9\), find \(f'(x)\).
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Answer#22
\(12x^3-7\).
Trap: Keeping the constant 9.
Product & Quotient Rules✓ Known #23
Product rule: if \(y=uv\), what is \(y’\)?
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Answer#23
\(y’=u’v+uv’\).
Trap: Writing \(u’v’\).
Product & Quotient Rules✓ Known #24
Quotient rule: if \(y=\dfrac{u}{v}\), what is \(y’\)?
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Answer#24
\(y’=\dfrac{u’v-uv’}{v^2}\).
Trap: Reversing the order in the numerator (a sign error).
Product & Quotient Rules✓ Known #25
Why does \((x^2)(x^3)\) not need the product rule?
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Answer#25
Simplify first: \(x^2x^3=x^5\), then use the power rule: \(5x^4\).
Trap: Using the product rule when simplifying is faster and safer.
Product & Quotient Rules✓ Known #26
Differentiate \(y=(x^2+1)x^3\).
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Answer#26
- \(y’=(2x)(x^3)+(x^2+1)(3x^2)\)
- \(=5x^4+3x^2\)
Trap: Differentiating both factors and multiplying.
Product & Quotient Rules✓ Known #27
Differentiate \(y=\dfrac{x^2+1}{x}\).
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Answer#27
\(y’=\dfrac{(2x)(x)-(x^2+1)(1)}{x^2}=\dfrac{x^2-1}{x^2}\).
Trap: Forgetting \(v^2\) in the denominator.
Chain Rule✓ Known #28
Chain rule: if \(y=f(g(x))\), what is \(\dfrac{dy}{dx}\)?
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Answer#28
\(f'(g(x))\cdot g'(x)\).
Trap: Forgetting the inner derivative.
Chain Rule✓ Known #29
Differentiate \(y=(3x-2)^5\).
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Answer#29
\(y’=5(3x-2)^4\cdot3=15(3x-2)^4\).
Trap: Missing the factor 3.
Chain Rule✓ Known #30
Differentiate \(y=\sqrt{2x+1}\).
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Answer#30
\(y’=\dfrac{1}{2\sqrt{2x+1}}\cdot2=\dfrac{1}{\sqrt{2x+1}}\).
Trap: Forgetting the inner derivative 2 (here it cancels the \(\frac12\)).
Chain Rule✓ Known #31
Differentiate \(y=\dfrac{1}{(x^2+1)^3}\).
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Answer#31
Rewrite as \((x^2+1)^{-3}\): \(y’=-3(x^2+1)^{-4}\cdot2x=\dfrac{-6x}{(x^2+1)^4}\).
Trap: Not rewriting with a negative exponent.
Chain Rule✓ Known #32
Differentiate \(y=(1-x)^7\).
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Answer#32
\(y’=-7(1-x)^6\).
Trap: Missing the negative from the inner derivative.
Chain Rule✓ Known #33
Differentiate \(y=(x^3-4x)^2\).
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Answer#33
\(y’=2(x^3-4x)(3x^2-4)\).
Trap: Differentiating only the outer function.
Chain Rule✓ Known #34
Differentiate \(y=(5-2x)^{-1}\).
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Answer#34
\(y’=-(5-2x)^{-2}\cdot(-2)=\dfrac{2}{(5-2x)^2}\).
Trap: A sign error from the double negative.
Chain Rule✓ Known #35
Differentiate \(y=\tan(x^2)\).
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Answer#35
\(y’=\sec^2(x^2)\cdot2x\).
Trap: Missing \(2x\).
Chain Rule✓ Known #36
Describe the chain rule in one sentence.
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Answer#36
Differentiate the outer function, keep the inside unchanged, then multiply by the derivative of the inside.
Trap: Changing the inside while differentiating the outside.
Chain Rule✓ Known #37
Set up the derivative of \(y=\left(\dfrac{x-1}{x+1}\right)^4\).
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Answer#37
\(y’=4\left(\dfrac{x-1}{x+1}\right)^3\cdot\dfrac{d}{dx}\left(\dfrac{x-1}{x+1}\right)\), and the last factor is \(\dfrac{2}{(x+1)^2}\).
Trap: Forgetting that the inner function needs the quotient rule.
Trig Derivatives✓ Known #38
\(\dfrac{d}{dx}(\sin x)=\,\)?
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Answer#38
\(\cos x\).
Trap: Answering \(-\cos x\).
Trig Derivatives✓ Known #39
\(\dfrac{d}{dx}(\cos x)=\,\)?
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Answer#39
\(-\sin x\).
Trap: Forgetting the negative.
Trig Derivatives✓ Known #40
\(\dfrac{d}{dx}(\tan x)=\,\)?
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Answer#40
\(\sec^2 x\).
Trap: Writing \(\sec x\).
Trig Derivatives✓ Known #41
Differentiate \(y=\sin(3x)\).
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Answer#41
\(y’=3\cos(3x)\).
Trap: Missing the 3.
Trig Derivatives✓ Known #42
Differentiate \(y=\cos(5x-1)\).
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Answer#42
\(y’=-5\sin(5x-1)\).
Trap: Losing the negative from cosine.
Trig Derivatives✓ Known #43
Differentiate \(y=\sin x\cos x\).
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Answer#43
Product rule: \(y’=\cos^2x-\sin^2x\) \((=\cos 2x)\).
Trap: Differentiating each factor and multiplying.
Trig Derivatives✓ Known #44
Differentiate \(y=\sec x\).
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Answer#44
\(y’=\sec x\tan x\).
Trap: Confusing it with \(\sec^2 x\).
Trig Derivatives✓ Known #45
Differentiate \(y=\csc x\).
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Answer#45
\(y’=-\csc x\cot x\).
Trap: Forgetting the negative.
Trig Derivatives✓ Known #46
Differentiate \(y=\cot x\).
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Answer#46
\(y’=-\csc^2 x\).
Trap: Missing the negative.
Exponential & Log Derivatives✓ Known #47
Differentiate \(y=e^x\).
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Answer#47
\(y’=e^x\).
Trap: Using the power rule on \(e^x\).
Exponential & Log Derivatives✓ Known #48
Differentiate \(y=a^x\) \((a>0,\ a\ne1)\).
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Answer#48
\(y’=a^x\ln a\).
Trap: Forgetting \(\ln a\).
Exponential & Log Derivatives✓ Known #49
Differentiate \(y=\ln x\).
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Answer#49
\(y’=\dfrac{1}{x}\) (for \(x>0\)).
Trap: Thinking \(\ln x\) is its own derivative.
Exponential & Log Derivatives✓ Known #50
Differentiate \(y=\ln(3x-2)\).
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Answer#50
\(y’=\dfrac{3}{3x-2}\).
Trap: Missing the inner derivative.
Exponential & Log Derivatives✓ Known #51
Differentiate \(y=e^{x^2}\).
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Answer#51
\(y’=2x\,e^{x^2}\).
Trap: Rewriting \(e^{x^2}\) as \(e^{2x}\).
Exponential & Log Derivatives✓ Known #52
Differentiate \(y=\ln\sqrt{x}\).
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Answer#52
\(\ln\sqrt{x}=\frac12\ln x\), so \(y’=\dfrac{1}{2x}\).
Trap: Forgetting the \(\frac12\).
Exponential & Log Derivatives✓ Known #53
Differentiate \(y=\dfrac{e^x}{x}\).
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Answer#53
Quotient rule: \(y’=\dfrac{xe^x-e^x}{x^2}=\dfrac{e^x(x-1)}{x^2}\).
Trap: A sign error in the numerator.
Implicit Differentiation✓ Known #54
Implicit differentiation: what is the key idea?
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Answer#54
Differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\), so \(\frac{d}{dx}(y)=\frac{dy}{dx}\).
Trap: Differentiating \(y\) as if it were a constant.
Implicit Differentiation✓ Known #55
Differentiate \(y^3\) with respect to \(x\).
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Answer#55
\(\dfrac{d}{dx}(y^3)=3y^2\dfrac{dy}{dx}\).
Trap: Missing the \(\frac{dy}{dx}\) factor.
Implicit Differentiation✓ Known #56
For \(x^2+y^2=25\), find \(\dfrac{dy}{dx}\).
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Answer#56
\(2x+2y\dfrac{dy}{dx}=0\Rightarrow\dfrac{dy}{dx}=-\dfrac{x}{y}\).
Trap: Writing \(2y\) instead of \(2y\frac{dy}{dx}\).
Implicit Differentiation✓ Known #57
For \(xy=10\), find \(\dfrac{dy}{dx}\).
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Answer#57
Product rule: \(x\dfrac{dy}{dx}+y=0\Rightarrow\dfrac{dy}{dx}=-\dfrac{y}{x}\).
Trap: Forgetting the product rule on \(xy\).
Implicit Differentiation✓ Known #58
For \(x^2+y^2=25\), what is the slope of the tangent line at \((3,4)\)?
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Answer#58
\(\dfrac{dy}{dx}=-\dfrac{x}{y}=-\dfrac34\).
Trap: Substituting only \(x\) and forgetting the \(y\)-value.
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