Derivatives - Page 6

Derivative of $\sqrt x$

Recall the power rule for taking derivatives of polynomials: $$\dfrac{d}{dx} x^n = nx^{n – 1}$$ The power rule works for all real values of $n$, not jus the integers. We may apply this rule to derive $\sqrt x = x^{\frac12}$. Using the rule above with $n =\frac12$ we get $$\dfrac{d}{dx} x^{\frac12} = \frac12 x^{-\frac12}$$ Usually,…

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Derivative of sin x, limit proof

The derivative of $\sin x$ is $\boxed{\cos x}$. We will use the limit definition of derivative to prove this. \[ \begin{align*} \frac{d}{dx} f(x) &= \lim_{h \to 0} \dfrac{f(x + h) – f(x)}h \end{align*} \] We will also use the trigonometric identity $\sin(a + b) = \cos a \sin b + \sin a \cos b$ (line…

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Derivative of a natural logarithm (ln x)

To find the derivative of $\ln x$, we will use implicit differentiation, which is a standard technique to find derivatives of inverse functions. Note that $\ln x$ is the inverse function of $e^x$, so it is natural for us to set $y = \ln x$. We know that $\dfrac{d}{dx} e^x = e^x$ and we will…

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Derivative of $e^{nx}$

The derivative of $e^{nx}$ is $\boxed{n\cdot e^{nx}}$. In this article, we will be exploring why. Recall that the Chain Rule states: \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x), \end{align*}\] where $f(x)$ and $g(x)$ are functions. In this case, we let $f(x) = e^x$ and $g(x) = nx$ so that $f(g(x)) = e^{g(x)} = e^{nx}$. The next…

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Derivative of $\cos x$

We will use two important facts to write a proof for the derivative of $\cos x$. Firstly, we will use the angle sum formula $$\cos(x+y) = \cos x\cos y – \sin x\sin y$$ This formula is part of precalculus and is common knowledge. The second formula we will use is $$\lim_{h\to0}\frac{\sin h} h = 1$$…

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Derivative of $\dfrac2x$

At first glance, it looks like we should use the quotient rule to differentiate $\dfrac2x$. Indeed, we do see a fraction. Recall the quotient rule. For functions $f(x)$ and $g(x)$, $$\dfrac d{dx}\left(\frac {f(x)}{g(x)}\right) = \frac{f'(x)g(x) – f(x)g'(x)}{g^2(x)}$$ If we let $f(x) = 2$ and $g(x) = x$, then $f'(x) = 0$ and $g'(x) = 1$.…

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Derivative of 1/x

To find the derivative of 1/x, we rewrite it as $x^{-1}$. We may apply the power rule $\dfrac{d}{dx} x^n = nx^{n – 1}$: \[ \begin{align*} \dfrac{d}{dx} x^{-1} &= (-1) \cdot x^{-1-1}\\ \dfrac{d}{dx} x^{-1} &= -x^{-2} \end{align*} \] We get the end result that $\boxed{\dfrac{d}{dx} \dfrac1x = \frac{d}{dx} x^{-1} = -\dfrac1{x^2}}$. In fact, this technique generalizes…

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Derivative of $\dfrac1{1+x}$

To find the derivative of $\dfrac{1}{1+x}$, we will apply the chain rule $$\dfrac{d}{dx}(f(g(x)) = f'(g(x))g'(x)$$ on the functions $f(x) = \dfrac1x$ and $g(x) = 1+x$. First, we need to find the derivatives of $f(x)$ and $g(x)$: $$f'(x) = \dfrac{d}{dx}(x^{-1}) = -x^{-2} = -\frac{1}{x^2}$$ and $$g'(x) = \dfrac{d}{dx}(x+1) = 1$$ We are now ready to apply…

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Proof of derivative of sin x

Understanding integrals and derivatives of trigonometric functions can be tough. For example, You may have been told that $\dfrac{d}{dx} \sin(x) = \cos(x)$, but not given a good explanation for it. In this article, we will show you with proof how to find the derivatives and integrals of $\cos x$ and $\sin x$ functions. Derivatives of…

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