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Derivative of ln(x^3)

We will be trying to find the derivative of $\ln(x^3)$. Using the Power Rule for logarithms, which tells us $\ln(x^n) = n \ln (x)$, we get $\ln(x^3) = 3\ln(x)$. Using the fact that $\dfrac{d}{dx} (\ln(x)) = \dfrac{1}{x}$, we have \[\begin{align*} \dfrac{d}{dx}(\ln(x^3)) = \dfrac{d}{dx}(3\ln(x)) = 3 \cdot \dfrac{d}{dx} (\ln (x)) = 3 \cdot \dfrac{1}{x} = \dfrac{3}{x}.…

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Derivative of ln(x^2)

We will be trying to find the derivative of $\ln(x^2)$. Recall the formula for the Chain Rule, which states that \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x). \end{align*}\] Using this formula, we let $f(x) = \ln(x)$ and $g(x) = x^2$ so that $f(g(x)) = \ln(g(x)) = \ln(x^2)$. To plug these values into the chain rule, we…

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Derivative of a^x

We will be finding the derivative of $a^x$ where $a$ is a constant. We will start by rewriting $a^x$ as: \[\begin{align*} a^x = (e^{\ln a})^x = e^{\ln a \cdot x}. \end{align*}\] Now, recall the formula $\dfrac{d}{dx}(e^{nx}) = e^{nx} \cdot n$. In the context of our problem, $n = \ln a$. More information on this formula…

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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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$\log_b(b^k)$ Rule

\[\begin{align*} \boxed{\log_{b}(b^k) = k} \end{align*}\] This rule says that if we take the log of a number that is to the $k$th power of the base $b$, the result is simply the exponent $k$. In other words: \[\begin{align*} \log_{b}(b^k) = k \end{align*}\] This rule works because the logarithm $\log_b(x)$ asks, “What power must we raise…

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Evaluating $\log_{10}(1000)$

What does $\log_{10}(1000) = X$ mean? We start with the expression: \[\begin{align*} \log_{10}(1000) = X \end{align*}\] This equation means that we are looking for a value of \(X\) such that: \[\begin{align*} 10^X = 1000 \end{align*}\] In other words, \(\log_{10}(1000)\) represents the power to which we must raise 10 to obtain 1000. Our goal is to…

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