Math Snap

STEP 1

1. We are given two functions: $f(x) = \frac{5}{\sqrt{x}}$ and $g(x) = 10 \sqrt{x}$.
2. We need to determine which function is an antiderivative of the other.

STEP 2

1. Find the antiderivative of $f(x)$.
2. Find the antiderivative of $g(x)$.
3. Compare the results to determine which function is an antiderivative of the other.

STEP 3

Find the antiderivative of $f(x) = \frac{5}{\sqrt{x}}$.
Rewrite $f(x)$ in terms of exponents:
$$f(x) = 5x^{-1/2}$$

STEP 4

Integrate $f(x) = 5x^{-1/2}$:
$$\int 5x^{-1/2} \, dx = 5 \cdot \frac{x^{1/2}}{1/2} + C$$ Simplify:
$$= 10x^{1/2} + C$$

STEP 5

Find the antiderivative of $g(x) = 10\sqrt{x}$.
Rewrite $g(x)$ in terms of exponents:
$$g(x) = 10x^{1/2}$$

STEP 6

Integrate $g(x) = 10x^{1/2}$:
$$\int 10x^{1/2} \, dx = 10 \cdot \frac{x^{3/2}}{3/2} + C$$ Simplify:
$$= \frac{20}{3}x^{3/2} + C$$

Compare the antiderivatives:
The antiderivative of $f(x) = \frac{5}{\sqrt{x}}$ is $10\sqrt{x} + C$, which matches $g(x)$.
The antiderivative of $g(x) = 10\sqrt{x}$ is $\frac{20}{3}x^{3/2} + C$, which does not match $f(x)$.
Therefore, $g(x) = 10\sqrt{x}$ is an antiderivative of $f(x) = \frac{5}{\sqrt{x}}$.
The function $g(x) = 10\sqrt{x}$ is an antiderivative of $f(x) = \frac{5}{\sqrt{x}}$.