Find the derivative of the following function. y=ln(2x^(6)) Answer: y^(')=

Identify function: Identify the function to differentiate. We have y = ln(2x^(6)). We need to find the derivative of this function with respect to x.Apply chain rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, the outer function is ln(u) and the inner function is u = 2x^(6).Differentiate outer function: Differentiate the outer function. The derivative of ln(u) with respect to u is 1/u. So, (d/du)(ln(u)) = 1/u.Differentiate inner function: Differentiate the inner function. The derivative of 2x^(6) with respect to x is 12x^(5). So, (d/dx)(2x^(6)) = 12x^(5).Apply chain rule with derivatives: Apply the chain rule using the derivatives from steps 3 and 4. y' = (d/du)(ln(u)) * (d/dx)(u) y' = (1/u) * (12x^(5)) y' = (1/(2x^(6))) * (12x^(5))Simplify expression: Simplify the expression. y' = 12x^(5) / (2x^(6)) y' = 12 / (2x) y' = 6/x

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Find the derivative of the following function.

y=ln(2x^(6))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\ln \left(2 x^{6}\right)$ $\newline$ Answer: $y^{\prime}=$

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Q. Find the derivative of the following function.

\newline

y=\ln \left(2 x^{6}\right)

\newline

Answer:

y^{\prime}=

Identify function: Identify the function to differentiate. $\newline$ We have $y = \ln(2x^{6})$ . We need to find the derivative of this function with respect to $x$ .
Apply chain rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, the outer function is $\ln(u)$ and the inner function is $u = 2x^{6}$ .
Differentiate outer function: Differentiate the outer function. $\newline$ The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$ . $\newline$ So, $\frac{d}{du}(\ln(u)) = \frac{1}{u}$ .
Differentiate inner function: Differentiate the inner function. $\newline$ The derivative of $2x^{6}$ with respect to $x$ is $12x^{5}$ . $\newline$ So, $(\frac{d}{dx})(2x^{6}) = 12x^{5}$ .
Apply chain rule with derivatives: Apply the chain rule using the derivatives from steps $3$ and $4$ . $\newline$ y' = $(\frac{d}{du})(\ln(u)) \cdot (\frac{d}{dx})(u)$ $\newline$ y' = $(\frac{1}{u}) \cdot (12x^{5})$ $\newline$ y' = $(\frac{1}{2x^{6}}) \cdot (12x^{5})$
Simplify expression: Simplify the expression. $\newline$ $y' = \frac{12x^{5}}{2x^{6}}$ $\newline$ $y' = \frac{12}{2x}$ $\newline$ $y' = \frac{6}{x}$