Derivatives of the Constants worksheet for calculus practice.

Let’s solve each derivative problem step by step. We’ll use basic rules of differentiation:

- Power Rule: If y = xⁿ, then dy/dx = n·xⁿ⁻¹
- Constant Multiple Rule: If y = c·f(x), then dy/dx = c·f’(x)
- Derivative of a constant is 0.
- For square roots and fractions with exponents, rewrite them using fractional exponents first.

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1) y = 2x³

Use power rule:
dy/dx = 2 · 3x² = 6x²

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2) y = 4√x

Rewrite √x as x^(1/2):
y = 4x^(1/2)

Now differentiate:
dy/dx = 4 · (1/2)x^(-1/2) = 2x^(-1/2) = 2 / √x

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3) y = x^(2/5)

Power rule:
dy/dx = (2/5)x^(2/5 - 1) = (2/5)x^(-3/5) = 2/(5x^(3/5))

You can also write it as: 2/(5∛(x³)) but usually we leave it in exponent form unless asked otherwise.

Actually, let’s keep it simple:
= (2/5) x^(-3/5) or 2/(5x^{3/5})

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4) y = 3 / √x

Rewrite: √x = x^(1/2), so 1/√x = x^(-1/2)
→ y = 3x^(-1/2)

Differentiate:
dy/dx = 3 · (-1/2)x^(-3/2) = -3/(2x^(3/2))

Or written with radical: -3/(2x√x)

But again, exponent form is fine: -3/(2x^{3/2})

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5) y = π⁴

π is a constant (~3.14), so π⁴ is just a number — a constant.

Derivative of any constant is 0

So dy/dx = 0

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6) y = 5x²

Power rule:
dy/dx = 5 · 2x = 10x

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7) y = x⁵

Power rule:
dy/dx = 5x⁴ → 5x⁴

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8) y = √(x³)

Rewrite: √(x³) = (x³)^(1/2) = x^(3/2)

Differentiate:
dy/dx = (3/2)x^(1/2) = (3/2)√x

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9) y = ∛x

Cube root of x = x^(1/3)

Differentiate:
dy/dx = (1/3)x^(-2/3) = 1/(3x^(2/3))

Or: 1/(3∛(x²))

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10) y = 2 / √(x³)

First, rewrite denominator: √(x³) = x^(3/2)
So y = 2 · x^(-3/2)

Differentiate:
dy/dx = 2 · (-3/2)x^(-5/2) = -3x^(-5/2) = -3 / x^(5/2)

Or: -3 / (x²√x)

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✔ Let me double-check all answers quickly:

1) 2x³ → 6x² ✔️
2) 4√x → 2/√x ✔️
3) x^(2/5) → (2/5)x^(-3/5) ✔️
4) 3/√x → -3/(2x^(3/2)) ✔️
5) π⁴ → 0 ✔️
6) 5x² → 10x ✔️
7) x⁵ → 5x⁴ ✔️
8) √(x³) → (3/2)√x ✔️
9) ∛x → 1/(3x^(2/3)) ✔️
10) 2/√(x³) → -3/x^(5/2) ✔️

All correct!

Final Answer:
1) 6x²
2) 2/√x
3) (2/5)x^(-3/5) or 2/(5x^{3/5})
4) -3/(2x^{3/2})
5) 0
6) 10x
7) 5x⁴
8) (3/2)√x
9) 1/(3x^{2/3})
10) -3/x^{5/2}