Differentiation: Composite, Implicit, and Inverse Functions — High School Calculus practice questions

Find d/dx[√(1 − x²)].

1. −x/√(1 − x²)
2. 1/(2√(1 − x²))
3. −2x/√(1 − x²)
4. x/√(1 − x²)

Answer: A — −x/√(1 − x²)

A) Correct — Chain rule: (1/2)(1 − x²)^(−1/2) · (−2x) = −x/√(1 − x²). B) Forgets to multiply by the derivative of the inside function (the chain rule). C) Forgets the factor 1/2 from the square root. D) Drops the negative sign from the derivative of −x².

Find d/dx[arcsin x].

1. 1/√(1 − x²)
2. 1/(1 + x²)
3. −1/√(1 − x²)
4. cos x

Answer: A — 1/√(1 − x²)

A) Correct — This is the standard inverse-sine derivative. B) Is the derivative of arctan x. C) Is the derivative of arccos x. D) Is the derivative of sin x, not arcsin x.

Find d/dx[e^(x²)].

1. e^(x²)
2. x² e^(x² − 1)
3. 2x e^(x²)
4. 2 e^(x²)

Answer: C — 2x e^(x²)

C) Correct — Chain rule: e^(x²) times the inner derivative 2x. A) Forgets to multiply by the derivative of the inside function (the chain rule). B) Applies the power rule to an exponential. D) Forgets the x in the inner derivative 2x.

Find d/dx[x² e^(3x)].

1. 2x e^(3x)
2. 2x e^(3x) + 3x² e^(3x)
3. 6x e^(3x)
4. 2x e^(3x) + x² e^(3x)

Answer: B — 2x e^(3x) + 3x² e^(3x)

B) Correct — Product rule, with the chain rule on e^(3x). A) Differentiates only the x² factor. C) Multiplies the two derivatives. D) Forgets the inner derivative 3 on e^(3x).

Find d/dx[2ˣ].

1. x·2^(x−1)
2. 2ˣ ln 2
3. 2ˣ
4. 2ˣ/ln 2

Answer: B — 2ˣ ln 2

B) Correct — For a constant base b, d/dx[bˣ] = bˣ ln b. A) Applies the power rule to an exponential. C) Treats 2ˣ like eˣ, missing the factor ln 2. D) Finds the antiderivative instead.

Implicitly differentiate x² + y² = 25. What is dy/dx?

1. x/y
2. −x/y
3. −y/x
4. 2x + 2y

Answer: B — −x/y

B) Correct — 2x + 2y·dy/dx = 0, so dy/dx = −x/y. A) Drops the negative sign when solving. C) Inverts the fraction. D) Differentiates but forgets the dy/dx on the y-term and never solves for it.

Find d/dx[e^(2x)].

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

Answer: A — 2e^(2x)

A) Correct — Chain rule: e^(2x) times the inner derivative 2. B) Forgets to multiply by the derivative of the inside function (the chain rule). C) Applies the power rule to an exponential. D) Finds the antiderivative instead.

Find d/dx[(3x² + 2)⁵].

1. 30x(3x² + 2)⁴
2. 5(3x² + 2)⁴
3. 6x(3x² + 2)⁴
4. 5(6x)⁴

Answer: A — 30x(3x² + 2)⁴

A) Correct — Chain rule: 5(3x² + 2)⁴ times the inner derivative 6x. B) Forgets to multiply by the derivative of the inside function (the chain rule). C) Multiplies by the inner derivative but drops the outer factor 5. D) Differentiates the inside and raises it to the 4th power.

Implicitly differentiate xy = 6. What is dy/dx?

1. −x/y
2. y/x
3. 0
4. −y/x

Answer: D — −y/x

D) Correct — Product rule: y + x·dy/dx = 0, so dy/dx = −y/x. A) Swaps x and y when solving. B) Drops the negative sign. C) Differentiates the right side (6) and forgets to apply the product rule to xy.

What is d/dx[arcsin x]?

1/√(1 − x²).

Find d/dx[xˣ] using log differentiation.

Let y = xˣ. ln y = x ln x. Diff: y'/y = ln x + 1 → y' = xˣ(ln x + 1).

What is logarithmic differentiation?

Take ln of both sides before differentiating, useful for products, quotients, or variable exponents.

What is the derivative of e^(3x)?

3e^(3x).