What would be the derivative of square roots? For example if I have $2 \sqrt{x}$ or $\sqrt{x}$.
I'm unsure how to find the derivative of these and include them especially in something like implicit.
What would be the derivative of square roots? For example if I have $2 \sqrt{x}$ or $\sqrt{x}$.
I'm unsure how to find the derivative of these and include them especially in something like implicit.
$\sqrt{x}$ Let $f(u)=u^{1/2}$ and $u=x$ That's $\frac{df}{du}=\frac{1}{2}u^{-1/2}$ and $\frac{du}{dx}=1$ But, by the chain rule $\frac{dy}{dx}=\frac{df}{du}•\frac{du}{dx} =\frac{1}{2}u^{-1/2} •1 =\frac{d}{dx}\sqrt{x}$ Finally $\frac{1}{2\sqrt{x}}$
Let $f(x) = \sqrt{x}$, then $f'(x) = \lim_{h \to 0} \dfrac{\sqrt{x+h} - \sqrt{x}}{h} = \lim_{h \to 0} \dfrac{\sqrt{x+h} - \sqrt{x}}{h} \times \dfrac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}} = \lim_{x \to 0} \dfrac{x+h-x}{h (\sqrt{x+h} + \sqrt{x})}\\ = \lim_{h \to 0} \dfrac{h}{h (\sqrt{x+h} + \sqrt{x})} = \lim_{h \to 0} \dfrac1{(\sqrt{x+h} + \sqrt{x})} = \dfrac1{2\sqrt{x}}$ In general, you can use the fact that if $f(x) = x^{t}$, then $f'(x) = tx^{t-1}$.
Taking $t=1/2$, gives us that $f'(x) = \dfrac12 x^{-1/2}$, which is the same as we obtained above.
Also, recall that $\dfrac{d (c f(x))}{dx} = c \dfrac{df(x)}{dx}$. Hence, you can pull out the constant and then differentiate it.
$\sqrt x=x^{1/2}$, so you just use the power rule: the derivative is $\frac12x^{-1/2}$.
The Power Rule says that $\frac{\mathrm{d}}{\mathrm{d}x}x^\alpha=\alpha x^{\alpha-1}$. Applying this to $\sqrt{x}=x^{\frac12}$ gives $ \begin{align} \frac{\mathrm{d}}{\mathrm{d}x}\sqrt{x} &=\frac{\mathrm{d}}{\mathrm{d}x}x^{\frac12}\\ &=\frac12x^{-\frac12}\\ &=\frac{1}{2\sqrt{x}}\tag{1} \end{align} $ However, if you are uncomfortable applying the Power Rule to a fractional power, consider applying implicit differentiation to $ \begin{align} y&=\sqrt{x}\\ y^2&=x\\ 2y\frac{\mathrm{d}y}{\mathrm{d}x}&=1\\ \frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{1}{2y}\\ &=\frac{1}{2\sqrt{x}}\tag{2} \end{align} $
Another possibility to find the derivative of $f(x)=\sqrt x$ is to use geometry. Imagine a square with side length $\sqrt x$. Then the area of the square is $x$. Now, let's extend the square on both sides by a small amount, $d\sqrt x$. The new area added to the square is: $dx=d\sqrt x * \sqrt x + d\sqrt x * \sqrt x + d\sqrt x^2.$
This is the sum of the sub-areas added on each side of the square (the orange areas in the picture above). The last term in the equation above is very small and can be neglected. Thus:
$dx=2*d\sqrt x * \sqrt x$
$\frac{dx}{d\sqrt x}=2 * \sqrt x$
$\frac{d\sqrt x}{dx}=\frac{1}{2*\sqrt x}$
(To go from the second step to the last, flip the fractions on both sides of the equation.)
Reference: Essence of Calculus, Chapter 3
Let $f(x) = \sqrt{x} = x^{1/2}$.
$f'(x) = \frac{1}{2} x ^{-1/2}$
$f'(x) = \frac{1}{2x^{1/2}} = \frac{1}{2\sqrt{x}}$
If you post the specific implicit differentiation problem, it may help. The general guideline of writing the square root as a fractional power and then using the power and chain rule appropriately should be fine however. Also, remember that you can simply pull out a constant when dealing with derivatives - see below.
If $g(x) = 2\sqrt{x} = 2x^{1/2}$. Then,
$g'(x) = 2\cdot\frac{1}{2}x^{-1/2}$
$g'(x) = \frac{1}{x^{1/2}} = \frac{1}{\sqrt{x}}$