AP Calc BC Differentiation Fundamentals — Questions and Answers
Question 1: Let f(x) = x^3. Which of the following limits equals f'(2)?
- lim(h→0) [(2+h)^3 − 2^3] / h, which equals 6
- lim(h→0) [(2+h)^3 − 2^3] / h, which equals 8
- lim(h→0) [(2+h)^3 − 2^3] / h, which equals 12 (Correct answer)
- lim(h→0) [(2+h)^3 − 2^3] / h, which equals 24
Correct answer: lim(h→0) [(2+h)^3 − 2^3] / h, which equals 12
Expanding (2+h)^3 − 8 gives 12h + 6h^2 + h^3. Dividing by h and letting h→0 gives 12, which matches f'(2) = 3(2)^2 = 12.
The derivative at a point is defined as f'(a) = lim(h→0) [f(a+h) − f(a)]/h. For f(x) = x^3 and a = 2, the numerator is (2+h)^3 − 8 = 12h + 6h^2 + h^3. Every term has a factor of h, so the quotient simplifies to 12 + 6h + h^2. As h→0 this approaches 12. The power rule gives the same result, 3x^2 evaluated at 2.
Question 2: If y = ln(cos(3x)), what is dy/dx?
- −3 tan(3x) (Correct answer)
- tan(3x)
- −tan(3x)
- −3 cot(3x)
Correct answer: −3 tan(3x)
Apply the chain rule twice: d/dx ln(u) = u'/u with u = cos(3x), and u' = −3 sin(3x). This gives −3 sin(3x)/cos(3x) = −3 tan(3x).
The function is a composition of three functions: ln, cos, and 3x. The derivative of ln(u) is u'/u. Here u = cos(3x), and its derivative is −sin(3x)·3 by the chain rule. Dividing, −3 sin(3x)/cos(3x) simplifies to −3 tan(3x). Leaving out the inner factor of 3 is the most common error.
Question 3: The curve x^2 + xy + y^2 = 7 passes through the point (1, 2). What is dy/dx at that point?
- 4/5
- −5/4
- −2/3
- −4/5 (Correct answer)
Correct answer: −4/5
Differentiating implicitly gives 2x + y + x(dy/dx) + 2y(dy/dx) = 0. At (1, 2) this becomes 4 + 5(dy/dx) = 0, so dy/dx = −4/5.
Differentiate each term with respect to x, remembering that y depends on x. The xy term needs the product rule, which gives y + x·y'. The y^2 term gives 2y·y'. So 2x + y + (x + 2y)y' = 0, which gives y' = −(2x + y)/(x + 2y). Substituting x = 1 and y = 2 gives −4/5.
Question 4: Let f(x) = x^3 + x + 1, which is one-to-one. If g is the inverse function of f, what is g'(3)?
- 1/3
- 1/4 (Correct answer)
- 4
- 3/4
Correct answer: 1/4
Since f(1) = 3, g(3) = 1. Then g'(3) = 1/f'(g(3)) = 1/f'(1) = 1/(3·1 + 1) = 1/4.
The inverse function derivative formula is (f⁻¹)'(b) = 1/f'(f⁻¹(b)). First solve f(x) = 3, which gives x = 1. Next, f'(x) = 3x^2 + 1, so f'(1) = 4. The reciprocal of that slope is 1/4. The error to watch for is evaluating f' at 3 instead of at the x-value that f maps to 3.
Question 5: What is the value of the derivative of f(x) = arctan(2x) at x = 1/2?
- 2
- 1/2
- 1 (Correct answer)
- 4/5
Correct answer: 1
d/dx arctan(2x) = 2/(1 + (2x)^2). At x = 1/2 this is 2/(1 + 1) = 1.
The derivative of arctan(u) is u'/(1 + u^2), and the chain rule supplies the factor u' = 2. So f'(x) = 2/(1 + 4x^2). Substituting x = 1/2 gives 4x^2 = 1, so f'(1/2) = 2/2 = 1. A common mistake is to drop the chain-rule factor of 2, which gives 1/2.
Question 6: If g(x) = x^2/(x + 1), what is g'(1)?
- 3/4 (Correct answer)
- 1/2
- 5/4
- 1/4
Correct answer: 3/4
By the quotient rule, g'(x) = [2x(x+1) − x^2(1)]/(x+1)^2. At x = 1 this is (4 − 1)/4 = 3/4.
The quotient rule states (u/v)' = (u'v − uv')/v^2. With u = x^2 and v = x + 1, the numerator is 2x(x+1) − x^2 = x^2 + 2x. At x = 1 the numerator is 3 and the denominator is (2)^2 = 4. So g'(1) = 3/4. Reversing the order of the subtraction in the numerator is the usual source of error.
Question 7: Let f(x) = x^2 for x ≤ 1 and f(x) = 2x − 1 for x > 1. Which statement is true at x = 1?
- f is not continuous at x = 1
- f is continuous but not differentiable at x = 1
- f is differentiable at x = 1 and f'(1) = 1
- f is differentiable at x = 1 and f'(1) = 2 (Correct answer)
Correct answer: f is differentiable at x = 1 and f'(1) = 2
Both pieces equal 1 at x = 1, so f is continuous. The left derivative is 2x = 2 and the right derivative is 2, so f'(1) = 2.
A function is differentiable at a point only if it is continuous there and the left and right derivatives agree. The left piece gives 1^2 = 1 and the right piece gives 2(1) − 1 = 1, so the pieces join. The derivative of x^2 at 1 is 2, and the derivative of 2x − 1 is the constant 2. The slopes match, so there is no corner and f'(1) = 2.
Question 8: What is the slope of the tangent line to the circle x^2 + y^2 = 25 at the point (4, −3)?
- −4/3
- 3/4
- 4/3 (Correct answer)
- −3/4
Correct answer: 4/3
Implicit differentiation gives 2x + 2y(dy/dx) = 0, so dy/dx = −x/y. At (4, −3) this is −4/(−3) = 4/3.
Differentiating x^2 + y^2 = 25 with respect to x gives 2x + 2y·y' = 0, so y' = −x/y. At (4, −3), y is negative, which makes the slope positive: −4/(−3) = 4/3. The radius to this point has slope −3/4, and the tangent is perpendicular to it. That confirms the answer, because the negative reciprocal of −3/4 is 4/3.
Question 9: What is d/dx [3^(x^2)]?
- x^2 · 3^(x^2 − 1)
- 2x · ln 3 · 3^(x^2) (Correct answer)
- 2x · 3^(x^2)
- ln 3 · 3^(x^2)
Correct answer: 2x · ln 3 · 3^(x^2)
For a constant base, d/dx a^u = a^u · ln a · u'. With a = 3 and u = x^2, the result is 3^(x^2) · ln 3 · 2x.
An exponential with constant base a has derivative a^u · ln(a) · u'. Here u = x^2, so u' = 2x. Combining these gives 2x · ln 3 · 3^(x^2). The power rule would only apply if the variable were in the base and the exponent were constant. Leaving out ln 3 or the inner factor 2x are the typical errors.
Question 10: What is d/dx [arcsin(x^2)]?
- 2x / √(1 − x^4) (Correct answer)
- 2x / √(1 − x^2)
- 1 / √(1 − x^4)
- 2x / (1 + x^4)
Correct answer: 2x / √(1 − x^4)
d/dx arcsin(u) = u'/√(1 − u^2). With u = x^2 this gives 2x/√(1 − x^4).
The derivative of arcsin(u) is u'/√(1 − u^2). Taking u = x^2, we have u' = 2x and u^2 = x^4. So the derivative is 2x/√(1 − x^4). Dropping the chain-rule factor gives choice C. Using the arctan form gives choice D. Writing x^2 in place of x^4 under the radical gives choice B.
