Find the derivative of the function. $$ f(x)=4 x^{-7 / 8}+8 x^{-1 / 2} $$ $$ f^{\prime}(x)= $$ Need Help? Read It Watch It

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To find the derivative of the function
$$ f(x) = 4x^{-7/8} + 8x^{-1/2}, $$
we can apply the power rule for differentiation. The power rule states that if $$ f(x) = ax^n $$, then
$$ f'(x) = a \cdot n \cdot x^{n-1}. $$

Let's differentiate each term separately.

1. **Differentiate $$ 4x^{-7/8} $$:**
   $$
   \frac{d}{dx} \left( 4x^{-7/8} \right) = 4 \cdot \left( -\frac{7}{8} \right) x^{-7/8 - 1} = -\frac{28}{8} x^{-15/8} = -\frac{7}{2} x^{-15/8}.
   $$

2. **Differentiate $$ 8x^{-1/2} $$:**
   $$
   \frac{d}{dx} \left( 8x^{-1/2} \right) = 8 \cdot \left( -\frac{1}{2} \right) x^{-1/2 - 1} = -4 x^{-3/2}.
   $$

3. **Combine the derivatives:**
   $$
   f'(x) = -\frac{7}{2} x^{-15/8} - 4x^{-3/2}.
   $$

This can also be written using fractional exponents in the denominator:
$$
f'(x) = -\frac{7}{2}\frac{1}{x^{15/8}} - \frac{4}{x^{3/2}}.
$$

**Final Answer:**
$$
f^{\prime}(x) = -\,\frac{7}{2}\,x^{-15/8}\; -\;4\,x^{-3/2}
$$
The derivative of the function $$ f(x) = 4x^{-7/8} + 8x^{-1/2} $$ is: $$ f^{\prime}(x) = -\frac{7}{2}x^{-15/8} - 4x^{-3/2} $$

Find the derivative of the function. \[ f(x)=4 x^{-7 / 8}+8 x^{-1 / 2} \] \( f^{\prime}(x)= \) Need Help? Read It Watch It

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