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

Studdy Solution

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms