To find the derivative of the given function [tex]\(36 \sqrt[9]{x^4} + 117 \sqrt[9]{x^{13}}\)[/tex], we start by rewriting the function in terms of fractional exponents.
The given function is:
[tex]\[ 36 \sqrt[9]{x^4} + 117 \sqrt[9]{x^{13}} \][/tex]
First, express each term with a fractional exponent:
[tex]\[ \sqrt[9]{x^4} = x^{4/9} \][/tex]
[tex]\[ \sqrt[9]{x^{13}} = x^{13/9} \][/tex]
Thus, the function can be rewritten as:
[tex]\[ 36 x^{4/9} + 117 x^{13/9} \][/tex]
Now, we will find the derivative of this function with respect to [tex]\( x \)[/tex].
### Step-by-Step Differentiation:
1. Differentiate [tex]\(36 x^{4/9}\)[/tex]:
[tex]\[ \frac{d}{dx} \left( 36 x^{4/9} \right) \][/tex]
Using the power rule of differentiation [tex]\(\frac{d}{dx} \left( x^n \right) = n x^{n-1}\)[/tex], we get:
[tex]\[ 36 \cdot \frac{4}{9} x^{4/9 - 1} \][/tex]
Simplify the exponent:
[tex]\[ 4/9 - 1 = 4/9 - 9/9 = -5/9 \][/tex]
Therefore:
[tex]\[ 36 \cdot \frac{4}{9} x^{-5/9} = 16 x^{-5/9} \][/tex]
2. Differentiate [tex]\(117 x^{13/9}\)[/tex]:
[tex]\[ \frac{d}{dx} \left( 117 x^{13/9} \right) \][/tex]
Again, using the power rule:
[tex]\[ 117 \cdot \frac{13}{9} x^{13/9 - 1} \][/tex]
Simplify the exponent:
[tex]\[ 13/9 - 1 = 13/9 - 9/9 = 4/9 \][/tex]
Therefore:
[tex]\[ 117 \cdot \frac{13}{9} x^{4/9} = 169 x^{4/9} \][/tex]
3. Combine the results:
[tex]\[ \frac{d}{dx} \left( 36 x^{4/9} + 117 x^{13/9} \right) = 16 x^{-5/9} + 169 x^{4/9} \][/tex]
### Writing the answer using radicals without negative exponents:
To rewrite the answer in terms of radicals without negative exponents, recall that:
[tex]\[ x^{-5/9} = \frac{1}{x^{5/9}} = \frac{1}{\sqrt[9]{x^5}} \][/tex]
[tex]\[ x^{4/9} = \sqrt[9]{x^4} \][/tex]
Thus, the derivative in terms of radicals is:
[tex]\[ 16 x^{-5/9} + 169 x^{4/9} = \frac{16}{\sqrt[9]{x^5}} + 169 \sqrt[9]{x^4} \][/tex]
### Final answers:
- Using fractional exponents:
[tex]\[ \boxed{16 x^{-5/9} + 169 x^{4/9}} \][/tex]
- Using radicals without negative exponents:
[tex]\[ \boxed{\frac{16}{\sqrt[9]{x^5}} + 169 \sqrt[9]{x^4}} \][/tex]
