November 21, 2024

Derivative of $\ln(x^3)$

In this article, we will be trying to show that: \[\begin{align*} \dfrac{d}{dx}(\ln(x^3)) = \boxed{\dfrac3x}. \end{align*}\] To start our proof, we will use the Power Rule for logarithms, which tells us $\ln(x^n) = n \ln (x)$. Using this rule, we get that $\ln(x^3) = 3\ln(x)$. Using the fact that $\dfrac{d}{dx} (\ln(x)) = \dfrac{1}{x}$, we have: \[\begin{align*}…

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Derivative of $\ln(x^2)$

In this article, we will be showing that: \[\begin{align*} \dfrac{d}{dx}(\ln(x^2)) = \boxed{\dfrac2x}. \end{align*}\] To prove this, we will use the Chain Rule. Recalling the formula for the Chain Rule, we find that the Chain Rule states that: \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x). \end{align*}\] Where $f(x)$ and $g(x)$ are differentiable functions. Using this formula, we…

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Derivative of -sin x

We will be finding the derivative of $-\sin x$. We first rewrite this expression as: \[\begin{align*} -\sin x = -1 \cdot \sin x. \end{align*}\] We can remove the $-1$ from the expression because it is a constant. Then, we recall that: \[\begin{align*} \dfrac{d}{dx}(\sin x) = \cos x. \end{align*}\] To learn about how we can prove…

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