Certainly! Let's differentiate the function [tex]\( F(x) = \frac{5}{6} x^{10} \)[/tex] step-by-step.
1. Identify the function and form: We are given the function [tex]\( F(x) = \frac{5}{6} x^{10} \)[/tex].
2. Apply the power rule: To differentiate a function of the form [tex]\( ax^n \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( n \)[/tex] is a nonzero integer, we use the power rule. The power rule states that [tex]\( \frac{d}{dx} (ax^n) = a \cdot n \cdot x^{n-1} \)[/tex].
3. Differentiate step-by-step:
a. We have [tex]\( a = \frac{5}{6} \)[/tex] and [tex]\( n = 10 \)[/tex].
b. According to the power rule, we multiply [tex]\( a \)[/tex] by [tex]\( n \)[/tex], and then multiply by [tex]\( x \)[/tex] raised to the power of [tex]\( n-1 \)[/tex].
4. Compute the derivative:
[tex]\[
F'(x) = \frac{d}{dx} \left( \frac{5}{6} x^{10} \right)
\][/tex]
Applying the power rule:
[tex]\[
F'(x) = \frac{5}{6} \cdot 10 \cdot x^{10-1}
\][/tex]
5. Simplify the expression:
[tex]\[
F'(x) = \frac{5}{6} \cdot 10 \cdot x^9 = \frac{50}{6} x^9
\][/tex]
Simplify the fraction [tex]\( \frac{50}{6} \)[/tex]:
[tex]\[
\frac{50}{6} = 8.33333333333333
\][/tex]
6. Final result:
[tex]\[
F'(x) = 8.33333333333333 \cdot x^9
\][/tex]
So, the derivative of the function [tex]\( F(x) = \frac{5}{6} x^{10} \)[/tex] is:
[tex]\[
F'(x) = 8.33333333333333 x^9
\][/tex]
