Find the derivative of the following function. y=ln(-2x^(3)) Answer: y^(')=

Apply Chain Rule: First, we need to apply the chain rule to differentiate the function y=ln(-2x^(3)). 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.Find Derivative of Inner Function: The outer function is the natural logarithm ln(u), and its derivative is 1/u. The inner function is -2x^3, and we will find its derivative next.Apply Chain Rule Again: The derivative of the inner function -2x^3 with respect to x is found by using the power rule, which states that the derivative of x^n is n*x^(n-1). So the derivative of -2x^3 is -2*3*x^(3-1) = -6x^2.Simplify Expression: Now we can apply the chain rule. The derivative of y with respect to x is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This gives us y' = (1/(-2x^3)) * (-6x^2).Cancel Common Terms: Simplify the expression by multiplying the two terms. This gives us y' = -6x^2 / (-2x^3).Simplify Constant Terms: We can simplify the expression further by canceling out the common terms. The x^2 in the numerator and one of the x's in the denominator cancel out, and the negatives cancel each other. This leaves us with y' = 6 / (2x).Finally, we can simplify the constant terms. 6 divided by 2 is 3, so the final simplified derivative is y' = 3/x.

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Find the derivative of the following function.

y=ln(-2x^(3))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\ln \left(-2 x^{3}\right)$ $\newline$ Answer: $y^{\prime}=$

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Calculus

Find derivatives using logarithmic differentiation

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Q. Find the derivative of the following function.

\newline

y=\ln \left(-2 x^{3}\right)

\newline

Answer:

y^{\prime}=

Apply Chain Rule: First, we need to apply the chain rule to differentiate the function $y=\ln(-2x^{3})$ . 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.
Find Derivative of Inner Function: The outer function is the natural logarithm $\ln(u)$ , and its derivative is $\frac{1}{u}$ . The inner function is $-2x^3$ , and we will find its derivative next.
Apply Chain Rule Again: The derivative of the inner function $-2x^3$ with respect to $x$ is found by using the power rule, which states that the derivative of $x^n$ is $n\cdot x^{(n-1)}$ . So the derivative of $-2x^3$ is $-2\cdot 3\cdot x^{(3-1)} = -6x^2$ .
Simplify Expression: Now we can apply the chain rule. The derivative of $y$ with respect to $x$ is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This gives us $y' = \frac{1}{-2x^3} \times (-6x^2)$ .
Cancel Common Terms: Simplify the expression by multiplying the two terms. This gives us $y' = \frac{-6x^2}{-2x^3}$ .
Simplify Constant Terms: We can simplify the expression further by canceling out the common terms. The $x^2$ in the numerator and one of the $x$ 's in the denominator cancel out, and the negatives cancel each other. This leaves us with $y' = \frac{6}{2x}$ .
Simplify Constant Terms: We can simplify the expression further by canceling out the common terms. The $x^2$ in the numerator and one of the $x$ 's in the denominator cancel out, and the negatives cancel each other. This leaves us with $y' = \frac{6}{2x}$ .Finally, we can simplify the constant terms. $6$ divided by $2$ is $3$ , so the final simplified derivative is $y' = \frac{3}{x}$ .