Find the derivative of the following function. y=ln(4x^(5)) Answer: y^(')=

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. For the function y = ln(4x^(5)), the outer function is ln(u) and the inner function is u = 4x^(5). First, we find the derivative of the outer function with respect to its argument u, which is 1/u. Then, we will find the derivative of the inner function with respect to x, which is the derivative of 4x^(5).Differentiate Inner Function: Differentiate the inner function 4x^(5). Using the power rule, the derivative of x^(n) with respect to x is n*x^(n-1). Therefore, the derivative of 4x^(5) with respect to x is 5*4x^(4) or 20x^(4).Combine Derivatives: Combine the derivatives using the chain rule. The derivative of y with respect to x is the derivative of the outer function times the derivative of the inner function. So, y' = (1/u) * (20x^(4)), where u = 4x^(5).Substitute Back: Substitute u back into the derivative. Replace u with 4x^(5) in the expression for y'. y' = (1/(4x^(5))) * (20x^(4))Simplify Expression: Simplify the expression. We can simplify the expression by canceling out x^(4) from the numerator and denominator. y' = (20x^(4))/(4x^(5)) = 20/4x = 5/x

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Find the derivative of the following function.

y=ln(4x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\ln \left(4 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives using logarithmic differentiation

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Q. Find the derivative of the following function.

\newline

y=\ln \left(4 x^{5}\right)

\newline

Answer:

y^{\prime}=

Apply Chain Rule: Apply the chain rule for differentiation. $\newline$ 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. $\newline$ For the function $y = \ln(4x^{5})$ , the outer function is $\ln(u)$ and the inner function is $u = 4x^{5}$ . $\newline$ First, we find the derivative of the outer function with respect to its argument $u$ , which is $1/u$ . $\newline$ Then, we will find the derivative of the inner function with respect to $x$ , which is the derivative of $4x^{5}$ .
Differentiate Inner Function: Differentiate the inner function $4x^{5}$ . Using the power rule, the derivative of $x^{n}$ with respect to $x$ is $n*x^{n-1}$ . Therefore, the derivative of $4x^{5}$ with respect to $x$ is $5*4x^{4}$ or $20x^{4}$ .
Combine Derivatives: Combine the derivatives using the chain rule. $\newline$ The derivative of $y$ with respect to $x$ is the derivative of the outer function times the derivative of the inner function. $\newline$ So, $y' = \frac{1}{u} \times (20x^{4})$ , where $u = 4x^{5}$ .
Substitute Back: Substitute $u$ back into the derivative. $\newline$ Replace $u$ with $4x^{5}$ in the expression for $y'$ . $\newline$ $y' = \frac{1}{4x^{5}} \cdot (20x^{4})$
Simplify Expression: Simplify the expression. $\newline$ We can simplify the expression by canceling out $x^{4}$ from the numerator and denominator. $\newline$ $y' = \frac{20x^{4}}{4x^{5}} = \frac{20}{4x} = \frac{5}{x}$