Differentiation Fundamentals Flashcards

Let f(x) = x^3. Which of the following limits equals f'(2)?

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.

If y = ln(cos(3x)), what is dy/dx?

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 curve x^2 + xy + y^2 = 7 passes through the point (1, 2). What is dy/dx at that point?

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.

Let f(x) = x^3 + x + 1, which is one-to-one. If g is the inverse function of f, what is g'(3)?

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.

What is the value of the derivative of f(x) = arctan(2x) at x = 1/2?

Answer: 1

d/dx arctan(2x) = 2/(1 + (2x)^2). At x = 1/2 this is 2/(1 + 1) = 1.

If g(x) = x^2/(x + 1), what is g'(1)?

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.

Let f(x) = x^2 for x ≤ 1 and f(x) = 2x − 1 for x > 1. Which statement is true at x = 1?

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.

What is the slope of the tangent line to the circle x^2 + y^2 = 25 at the point (4, −3)?

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.

What is d/dx [3^(x^2)]?

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.

What is d/dx [arcsin(x^2)]?

Answer: 2x / √(1 − x^4)

d/dx arcsin(u) = u'/√(1 − u^2). With u = x^2 this gives 2x/√(1 − x^4).