Solved on Jan 23, 2024

Find the derivative of y=log8(9xx5)y=\log_{8}(9-x-x^{5}).

STEP 1

Assumptions
1. We are given the function y=log8(9xx5)y=\log _{8}\left(9-x-x^{5}\right).
2. We need to find the derivative of yy with respect to xx, denoted as yy'.
3. We will use the chain rule and the logarithmic differentiation rule to find yy'.

STEP 2

The derivative of a logarithm with base aa is given by the formula:
ddxloga(u(x))=u(x)u(x)ln(a)\frac{d}{dx} \log_a(u(x)) = \frac{u'(x)}{u(x) \ln(a)}
where u(x)u(x) is a differentiable function of xx.

STEP 3

Identify u(x)u(x) in the given function:
u(x)=9xx5u(x) = 9 - x - x^5

STEP 4

Find the derivative of u(x)u(x) with respect to xx, denoted as u(x)u'(x):
u(x)=ddx(9xx5)u'(x) = \frac{d}{dx}(9 - x - x^5)

STEP 5

Differentiate each term of u(x)u(x) separately:
u(x)=ddx(9)ddx(x)ddx(x5)u'(x) = \frac{d}{dx}(9) - \frac{d}{dx}(x) - \frac{d}{dx}(x^5)

STEP 6

Calculate the derivatives:
u(x)=015x4u'(x) = 0 - 1 - 5x^4

STEP 7

Simplify the expression for u(x)u'(x):
u(x)=15x4u'(x) = -1 - 5x^4

STEP 8

Now apply the logarithmic differentiation rule to find yy':
y=u(x)u(x)ln(8)y' = \frac{u'(x)}{u(x) \ln(8)}

STEP 9

Substitute u(x)u(x) and u(x)u'(x) into the formula for yy':
y=15x4(9xx5)ln(8)y' = \frac{-1 - 5x^4}{(9 - x - x^5) \ln(8)}

STEP 10

Since the derivative should be expressed as a positive value, we can multiply the numerator and denominator by -1 to get:
y=1+5x4(x5+x9)ln(8)y' = \frac{1 + 5x^4}{(x^5 + x - 9) \ln(8)}

STEP 11

Now we can match the expression for yy' with the given options. The correct derivative is:
y=5x4+1(x5+x9)ln(8)y' = \frac{5x^4 + 1}{(x^5 + x - 9) \ln(8)}
This matches the first option given: 5x4+1ln8(x5+x9)\frac{5 x^{4}+1}{\ln 8\left(x^{5}+x-9\right)}

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