Find the derivative of h(x). h(x) = ln(x^2) h′(x) = ______

Identify outer and inner functions: Let's use the chain rule to differentiate h(x) = ln(x^2). 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. First, we identify the outer function as ln(u) where u = x^2, and the inner function as u = x^2. The derivative of ln(u) with respect to u is 1/u, and the derivative of u = x^2 with respect to x is 2x. Now we apply the chain rule: h′(x) = (1/u) * (du/dx) = (1/x^2) * (2x).Apply the chain rule: Simplify the expression for h′(x) by multiplying (1/x^2) by (2x) to get h′(x) = 2x/x^2.Simplify the expression: Further simplify the expression by canceling out one x from the numerator and the denominator to get h′(x) = 2/x.

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Find derivatives using the quotient rule I

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Q. Find the derivative of

h(x)

\newline

h(x) = \ln(x^2)

\newline

h'(x) =

______

Identify outer and inner functions: Let's use the chain rule to differentiate $h(x) = \ln(x^2)$ . 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. $\newline$ First, we identify the outer function as $\ln(u)$ where $u = x^2$ , and the inner function as $u = x^2$ . $\newline$ The derivative of $\ln(u)$ with respect to $u$ is $1/u$ , and the derivative of $u = x^2$ with respect to $x$ is $2x$ . $\newline$ Now we apply the chain rule: $\ln(u)$ $0$ .
Apply the chain rule: Simplify the expression for $h'(x)$ by multiplying $(1/x^2)$ by $(2x)$ to get $h'(x) = 2x/x^2$ .
Simplify the expression: Further simplify the expression by canceling out one $x$ from the numerator and the denominator to get $h'(x) = \frac{2}{x}$ .