Derivative (power Rule) online exercise for - Free Printable

Let’s solve each problem step by step. We’re using the Power Rule for derivatives, which says:

> If \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \)

We’ll also use basic algebra to rewrite functions so we can apply the power rule easily.

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Problem 1:

\( f(x) = \frac{3}{\sqrt{x}} + \frac{5}{x^2} \)

First, rewrite with exponents:

- \( \frac{3}{\sqrt{x}} = 3x^{-1/2} \)
- \( \frac{5}{x^2} = 5x^{-2} \)

Now differentiate term by term:

- Derivative of \( 3x^{-1/2} \):
\( 3 \cdot (-\frac{1}{2}) x^{-3/2} = -\frac{3}{2} x^{-3/2} = -\frac{3}{2\sqrt{x^3}} \)

- Derivative of \( 5x^{-2} \):
\( 5 \cdot (-2) x^{-3} = -10x^{-3} = -\frac{10}{x^3} \)

So,
\( f'(x) = -\frac{3}{2\sqrt{x^3}} - \frac{10}{x^3} \)

✔ This matches the second derivative on the right side.

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Problem 2:

\( f(x) = (x + 1)(3x^2 - 4) \)

We could use the product rule, but it’s easier to expand first:

Multiply out:

\( f(x) = x(3x^2 - 4) + 1(3x^2 - 4) = 3x^3 - 4x + 3x^2 - 4 \)

Rewrite in order:
\( f(x) = 3x^3 + 3x^2 - 4x - 4 \)

Now differentiate term by term:

- \( 3x^3 → 9x^2 \)
- \( 3x^2 → 6x \)
- \( -4x → -4 \)
- \( -4 → 0 \)

So,
\( f'(x) = 9x^2 + 6x - 4 \)

✔ This matches the third derivative on the right side.

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Problem 3:

\( f(x) = \frac{4x^2 - x + 3}{\sqrt{x}} \)

Rewrite denominator as exponent: \( \sqrt{x} = x^{1/2} \)

Split into separate terms:

\( f(x) = \frac{4x^2}{x^{1/2}} - \frac{x}{x^{1/2}} + \frac{3}{x^{1/2}} \)

Simplify each term using exponent rules (\( x^a / x^b = x^{a-b} \)):

- \( 4x^{2 - 1/2} = 4x^{3/2} \)
- \( -x^{1 - 1/2} = -x^{1/2} \)
- \( 3x^{-1/2} \)

So,
\( f(x) = 4x^{3/2} - x^{1/2} + 3x^{-1/2} \)

Now differentiate:

- \( 4x^{3/2} → 4 \cdot \frac{3}{2} x^{1/2} = 6x^{1/2} = 6\sqrt{x} \)
- \( -x^{1/2} → -\frac{1}{2} x^{-1/2} = -\frac{1}{2\sqrt{x}} \)
- \( 3x^{-1/2} → 3 \cdot (-\frac{1}{2}) x^{-3/2} = -\frac{3}{2} x^{-3/2} = -\frac{3}{2\sqrt{x^3}} \)

So,
\( f'(x) = 6\sqrt{x} - \frac{1}{2\sqrt{x}} - \frac{3}{2\sqrt{x^3}} \)

✔ This matches the first derivative on the right side.

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Final Answer:
Match each function to its correct derivative:

- \( f(x) = \frac{3}{\sqrt{x}} + \frac{5}{x^2} \) → \( f'(x) = -\frac{3}{2\sqrt{x^3}} - \frac{10}{x^3} \)
- \( f(x) = (x + 1)(3x^2 - 4) \) → \( f'(x) = 9x^2 + 6x - 4 \)
- \( f(x) = \frac{4x^2 - x + 3}{\sqrt{x}} \) → \( f'(x) = 6\sqrt{x} - \frac{1}{2\sqrt{x}} - \frac{3}{2\sqrt{x^3}} \)