Berikut adalah contoh soal bahasa Inggris tentang derivative (turunan) beserta jawaban dan penjelasannya, cocok untuk latihan matematika atau kalkulus tingkat SMA/SMK/early college:

1. Basic Derivative

Question:
Find the derivative of the function:$$f(x) = 3x^4 – 5x^2 + 7$$f(x)=3×4−5×2+7

Answer:$$f'(x) = 12x^3 – 10x$$f′(x)=12×3−10x

Explanation:
Use the power rule: $\frac{d}{dx}[x^n] = nx^{n-1}$dxd​[xn]=nxn−1.

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2. Derivative of a Trigonometric Function

Question:
Find $\frac{d}{dx}[\sin(x) + 2\cos(x)]$dxd​[sin(x)+2cos(x)]

Answer:$$\frac{d}{dx}[\sin(x) + 2\cos(x)] = \cos(x) – 2\sin(x)$$dxd​[sin(x)+2cos(x)]=cos(x)−2sin(x)

Explanation:
Derivative rules: $\frac{d}{dx} \sin(x) = \cos(x)$dxd​sin(x)=cos(x), $\frac{d}{dx} \cos(x) = -\sin(x)$dxd​cos(x)=−sin(x).

3. Derivative Using the Product Rule

Question:
Find the derivative of $y = x^2 \cdot e^x$y=x2⋅ex

Answer:$$y’ = 2x e^x + x^2 e^x = e^x (2x + x^2)$$y′=2xex+x2ex=ex(2x+x2)

Explanation:
Product rule: $\frac{d}{dx}[u \cdot v] = u’v + uv’$dxd​[u⋅v]=u′v+uv′, here $u = x^2$u=x2, $v = e^x$v=ex.

4. Derivative Using the Quotient Rule

Question:
Find the derivative of $f(x) = \frac{x^2 + 1}{x}$f(x)=xx2+1​

Answer:$$f'(x) = \frac{(2x)(x) – (x^2+1)(1)}{x^2} = \frac{2x^2 – x^2 – 1}{x^2} = \frac{x^2 – 1}{x^2} = 1 – \frac{1}{x^2}$$f′(x)=x2(2x)(x)−(x2+1)(1)​=x22x2−x2−1​=x2x2−1​=1−x21​

Explanation:
Quotient rule: $\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u’v – uv’}{v^2}$dxd​[vu​]=v2u′v−uv′​.

5. Chain Rule Derivative

Question:
Find the derivative of $y = \sqrt{3x^2 + 1}$y=3×2+1​

Answer:$$y’ = \frac{1}{2\sqrt{3x^2 + 1}} \cdot 6x = \frac{6x}{2\sqrt{3x^2 + 1}} = \frac{3x}{\sqrt{3x^2 + 1}}$$y′=23×2+1​1​⋅6x=23×2+1​6x​=3×2+1​3x​

Explanation:
Chain rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$dxd​[f(g(x))]=f′(g(x))⋅g′(x).

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6. Higher-Order Derivative

Question:
Find the second derivative of $f(x) = x^3 – 4x^2 + x$f(x)=x3−4×2+x

Answer:$$f'(x) = 3x^2 – 8x + 1$$f′(x)=3×2−8x+1$$f”(x) = 6x – 8$$f′′(x)=6x−8

Explanation:
Differentiate twice: first derivative, then differentiate again to get the second derivative.

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