Question Find the derivative of $y=\ln \left[x\left(x^{5}-2\right)^{10}\right]$ using logarithm properties.

Studdy Solution

STEP 1

Assumptions
1. We are given the function $y=\ln \left[x\left(x^{5}-2\right)^{10}\right]$ .
2. We need to find the derivative of $y$ with respect to $x$ .
3. We will use the properties of logarithms and the rules of differentiation to solve the problem.

STEP 2

Apply the logarithm properties to simplify the expression before differentiating. We can use the property that $\ln(a^b) = b\ln(a)$ and the property that $\ln(ab) = \ln(a) + \ln(b)$ .
$y = \ln \left[x\left(x^{5}-2\right)^{10}\right] = \ln(x) + \ln\left[\left(x^{5}-2\right)^{10}\right]$

STEP 3

Apply the logarithm power rule to the second term.
$y = \ln(x) + 10\ln\left(x^{5}-2\right)$

STEP 4

Now we will differentiate $y$ with respect to $x$ . We will use the chain rule and the power rule for differentiation.
$\frac{dy}{dx} = \frac{d}{dx}[\ln(x)] + 10\frac{d}{dx}[\ln\left(x^{5}-2\right)]$

STEP 5

Differentiate the first term using the derivative of the natural logarithm function, which is $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$ .
$\frac{dy}{dx} = \frac{1}{x} + 10\frac{d}{dx}[\ln\left(x^{5}-2\right)]$

STEP 6

Differentiate the second term using the chain rule. The derivative of $\ln(u)$ with respect to $x$ is $\frac{1}{u}\frac{du}{dx}$ , where $u = x^{5}-2$ .
$\frac{dy}{dx} = \frac{1}{x} + 10\left(\frac{1}{x^{5}-2}\frac{d}{dx}[x^{5}-2]\right)$

STEP 7

Differentiate the expression $x^{5}-2$ with respect to $x$ using the power rule, which states that $\frac{d}{dx}[x^n] = nx^{n-1}$ .
$\frac{d}{dx}[x^{5}-2] = 5x^{4}$

STEP 8

Substitute the derivative from STEP_7 into the equation from STEP_6.
$\frac{dy}{dx} = \frac{1}{x} + 10\left(\frac{1}{x^{5}-2}\cdot 5x^{4}\right)$

STEP 9

Simplify the expression by multiplying the constants and combining terms.
$\frac{dy}{dx} = \frac{1}{x} + \frac{50x^{4}}{x^{5}-2}$

STEP 10

The derivative of the function $y=\ln \left[x\left(x^{5}-2\right)^{10}\right]$ with respect to $x$ is:
$\frac{dy}{dx} = \frac{1}{x} + \frac{50x^{4}}{x^{5}-2}$
This is the final answer.

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