Answer :
To find the derivative of the function [tex]\( f(x) = 4 \cdot \sqrt[7]{x^2} \)[/tex], let's work through the process step-by-step.
1. Rewrite the Function:
First, we rewrite the function using exponent notation instead of the radical form:
[tex]\[ f(x) = 4 \cdot \left( x^2 \right)^{\frac{1}{7}} \][/tex]
2. Apply the Chain Rule:
To differentiate [tex]\( f(x) \)[/tex], we will apply the chain rule. The chain rule states that if you have a composite function [tex]\( h(x) = g(u(x)) \)[/tex], then the derivative is given by [tex]\( h'(x) = g'(u(x)) \cdot u'(x) \)[/tex].
3. Differentiate the Outer Function:
Let [tex]\( u = x^2 \)[/tex]. Then [tex]\( f(x) = 4 \cdot u^{\frac{1}{7}} \)[/tex].
The derivative of [tex]\( u^{\frac{1}{7}} \)[/tex] with respect to [tex]\( u \)[/tex] is:
[tex]\[ \frac{d}{du} \left( u^{\frac{1}{7}} \right) = \frac{1}{7} u^{-\frac{6}{7}} \][/tex]
4. Differentiate the Inner Function:
Next, we differentiate [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{du}{dx} = \frac{d}{dx} \left( x^2 \right) = 2x \][/tex]
5. Apply the Chain Rule:
Combining these derivatives using the chain rule, we get:
[tex]\[ \frac{d}{dx} \left[ 4 \cdot \left( x^2 \right)^{\frac{1}{7}} \right] = 4 \cdot \left( \frac{1}{7} \left( x^2 \right)^{-\frac{6}{7}} \right) \cdot 2x \][/tex]
6. Simplify the Expression:
Let’s simplify the expression:
[tex]\[ \frac{d}{dx} \left[ 4 \cdot \left( x^2 \right)^{\frac{1}{7}} \right] = 4 \cdot \frac{1}{7} \cdot \left( x^2 \right)^{-\frac{6}{7}} \cdot 2x \][/tex]
Combine the constants:
[tex]\[ = \frac{4 \cdot 2}{7} \cdot \left( x^2 \right)^{-\frac{6}{7}} \cdot x \][/tex]
[tex]\[ = \frac{8}{7} \cdot \left( x^2 \right)^{-\frac{6}{7}} \cdot x \][/tex]
7. Rewrite in Simplified Form:
Since [tex]\(\left( x^2 \right)^{-\frac{6}{7}} \cdot x = \left( x^2 \right)^{-\frac{6}{7}} \cdot x^1 \)[/tex]:
[tex]\[ = \frac{8}{7} \cdot x \cdot \left( x^2 \right)^{-\frac{6}{7}} \][/tex]
Rewriting [tex]\(\left( x^2 \right)^{-\frac{6}{7}}\)[/tex] as [tex]\(\left( x^2 \right)^{\frac{1}{7} - 1}\)[/tex] gives:
[tex]\[ = \frac{8}{7} \cdot x \cdot \left( x^2 \right)^{-\frac{6}{7}} \][/tex]
Simplifying the exponent:
[tex]\[ = \frac{8}{7} \cdot \left( x \cdot x^{- \frac{12}{7}} \right) \][/tex]
[tex]\[ = \frac{8}{7} \cdot x^{1 - \frac{12}{7}} \][/tex]
[tex]\[ = \frac{8}{7} \cdot x^{\frac{7}{7} - \frac{12}{7}} \][/tex]
[tex]\[ = \frac{8}{7} \cdot x^{-\frac{5}{7}} \][/tex]
Therefore, the derivative of the function [tex]\( f(x) = 4 \cdot \sqrt[7]{x^2} \)[/tex] is:
[tex]\[ f'(x) = \frac{8}{7} x^{-\frac{5}{7}} \][/tex]
In fraction form, this is equivalent to:
[tex]\[ f'(x) = \frac{8}{7} \cdot \frac{1}{x^{\frac{5}{7}}} \][/tex]
Let's express it in a simplified way, matching the given result:
[tex]\[ f'(x) = 1.14285714285714 \cdot \frac{(x^2)^{0.142857142857143}}{x} \][/tex]
1. Rewrite the Function:
First, we rewrite the function using exponent notation instead of the radical form:
[tex]\[ f(x) = 4 \cdot \left( x^2 \right)^{\frac{1}{7}} \][/tex]
2. Apply the Chain Rule:
To differentiate [tex]\( f(x) \)[/tex], we will apply the chain rule. The chain rule states that if you have a composite function [tex]\( h(x) = g(u(x)) \)[/tex], then the derivative is given by [tex]\( h'(x) = g'(u(x)) \cdot u'(x) \)[/tex].
3. Differentiate the Outer Function:
Let [tex]\( u = x^2 \)[/tex]. Then [tex]\( f(x) = 4 \cdot u^{\frac{1}{7}} \)[/tex].
The derivative of [tex]\( u^{\frac{1}{7}} \)[/tex] with respect to [tex]\( u \)[/tex] is:
[tex]\[ \frac{d}{du} \left( u^{\frac{1}{7}} \right) = \frac{1}{7} u^{-\frac{6}{7}} \][/tex]
4. Differentiate the Inner Function:
Next, we differentiate [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{du}{dx} = \frac{d}{dx} \left( x^2 \right) = 2x \][/tex]
5. Apply the Chain Rule:
Combining these derivatives using the chain rule, we get:
[tex]\[ \frac{d}{dx} \left[ 4 \cdot \left( x^2 \right)^{\frac{1}{7}} \right] = 4 \cdot \left( \frac{1}{7} \left( x^2 \right)^{-\frac{6}{7}} \right) \cdot 2x \][/tex]
6. Simplify the Expression:
Let’s simplify the expression:
[tex]\[ \frac{d}{dx} \left[ 4 \cdot \left( x^2 \right)^{\frac{1}{7}} \right] = 4 \cdot \frac{1}{7} \cdot \left( x^2 \right)^{-\frac{6}{7}} \cdot 2x \][/tex]
Combine the constants:
[tex]\[ = \frac{4 \cdot 2}{7} \cdot \left( x^2 \right)^{-\frac{6}{7}} \cdot x \][/tex]
[tex]\[ = \frac{8}{7} \cdot \left( x^2 \right)^{-\frac{6}{7}} \cdot x \][/tex]
7. Rewrite in Simplified Form:
Since [tex]\(\left( x^2 \right)^{-\frac{6}{7}} \cdot x = \left( x^2 \right)^{-\frac{6}{7}} \cdot x^1 \)[/tex]:
[tex]\[ = \frac{8}{7} \cdot x \cdot \left( x^2 \right)^{-\frac{6}{7}} \][/tex]
Rewriting [tex]\(\left( x^2 \right)^{-\frac{6}{7}}\)[/tex] as [tex]\(\left( x^2 \right)^{\frac{1}{7} - 1}\)[/tex] gives:
[tex]\[ = \frac{8}{7} \cdot x \cdot \left( x^2 \right)^{-\frac{6}{7}} \][/tex]
Simplifying the exponent:
[tex]\[ = \frac{8}{7} \cdot \left( x \cdot x^{- \frac{12}{7}} \right) \][/tex]
[tex]\[ = \frac{8}{7} \cdot x^{1 - \frac{12}{7}} \][/tex]
[tex]\[ = \frac{8}{7} \cdot x^{\frac{7}{7} - \frac{12}{7}} \][/tex]
[tex]\[ = \frac{8}{7} \cdot x^{-\frac{5}{7}} \][/tex]
Therefore, the derivative of the function [tex]\( f(x) = 4 \cdot \sqrt[7]{x^2} \)[/tex] is:
[tex]\[ f'(x) = \frac{8}{7} x^{-\frac{5}{7}} \][/tex]
In fraction form, this is equivalent to:
[tex]\[ f'(x) = \frac{8}{7} \cdot \frac{1}{x^{\frac{5}{7}}} \][/tex]
Let's express it in a simplified way, matching the given result:
[tex]\[ f'(x) = 1.14285714285714 \cdot \frac{(x^2)^{0.142857142857143}}{x} \][/tex]
