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Once upon a time, physics was simple and beautiful. Everything we knew about the universe could be derived from a small set of brief, elegant expressions called "Lagrangians." If you wanted to predict how a particle would behave in an electric or magnetic field, for example, you would start with this Lagrangian (from classical electrodynamics):[math]\epsilon_0 \cdot \frac{||\mathbf{E}||^2 - c^2||\mathbf{B}||^2}{2} - \rho \phi + \mathbf{j} \cdot \mathbf{A}[/math]where [math]\mathbf{E}[/math] is the electric field, [math]\mathbf{B}[/math] is the magnetic field, [math]\rho[/math] is the charge density, and [math]\mathbf{j}[/math] is the current density vector. You would plug this expression into the Euler-Lagrange equation, which minimizes the integral of the Lagrangian (also called the "action"). And just a few mathematical steps later, voilà! You had a complete description of the particle's motion.If you wanted to know how a particle would behave in a gravitational field, well, then, you used this Lagrangian (from general relativity):[math]\frac{1}{16\pi}(R-2\Lambda) \sqrt{-g}[/math]Solving the resulting equation was fairly complicated, but at least the Lagrangian itself was simple and easy to understand. And to make things even easier, you had the perfectly respectable option of ignoring the [math]\Lambda[/math]. Everyone knew that the cosmological constant was just a blunder.Then came the Standard Model. Its Lagrangian looks like this:[math]-\frac{1}{2}\partial_{\nu}g^{a}_{\mu}\partial_{\nu}g^{a}_{\mu} -g_{s}f^{abc}\partial_{\mu}g^{a}_{\nu}g^{b}_{\mu}g^{c}_{\nu} -\frac{1}{4}g^{2}_{s}f^{abc}f^{ade}g^{b}_{\mu}g^{c}_{\nu}g^{d}_{\mu}g^{e}_{\nu}[/math][math]+\frac{1}{2}ig^{2}_{s}(\bar{q}^{\sigma}_{i}\gamma^{\mu}q^{\sigma}_{j})g^{a}_{\mu} +\bar{G}^{a}\partial^{2}G^{a}+g_{s}f^{abc}\partial_{\mu}\bar{G}^{a}G^{b}g^{c}_{\mu} -\partial_{\nu}W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-M^{2}W^{+}_{\mu}W^{-}_{\mu}[/math][math]-\frac{1}{2}\partial_{\nu}Z^{0}_{\mu}\partial_{\nu}Z^{0}_{\mu}-\frac{1}{2c^{2}_{w}} M^{2}Z^{0}_{\mu}Z^{0}_{\mu} -\frac{1}{2}\partial_{\mu}A_{\nu}\partial_{\mu}A_{\nu}[/math][math]-\frac{1}{2}\partial_{\mu}H\partial_{\mu}H-\frac{1}{2}m^{2}_{h}H^{2} -\partial_{\mu}\phi^{+}\partial_{\mu}\phi^{-}-M^{2}\phi^{+}\phi^{-} -\frac{1}{2}\partial_{\mu}\phi^{0}\partial_{\mu}\phi^{0}-\frac{1}{2c^{2}_{w}}M\phi^{0}\phi^{0}[/math][math]-\beta_{h}[\frac{2M^{2}}{g^{2}}+\frac{2M}{g}H+\frac{1}{2}(H^{2}+\phi^{0}\phi^{0}+2\phi^{+}\phi^{-})] +\frac{2M^{4}}{g^{2}}\alpha_{h}[/math][math]-igc_{w}[\partial_{\nu}Z^{0}_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu}) -Z^{0}_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu}) +Z^{0}_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})][/math][math]-igs_{w}[\partial_{\nu}A_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu}) -A_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu}) +A_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})][/math][math]-\frac{1}{2}g^{2}W^{+}_{\mu}W^{-}_{\mu}W^{+}_{\nu}W^{-}_{\nu}+\frac{1}{2}g^{2}W^{+}_{\mu}W^{-}_{\nu}W^{+}_{\mu}W^{-}_{\nu}[/math][math]+g^2c^{2}_{w}(Z^{0}_{\mu}W^{+}_{\mu}Z^{0}_{\nu}W^{-}_{\nu}-Z^{0}_{\mu}Z^{0}_{\mu}W^{+}_{\nu} W^{-}_{\nu})[/math][math]+g^2s^{2}_{w}(A_{\mu}W^{+}_{\mu}A_{\nu}W^{-}_{\nu}-A_{\mu}A_{\mu}W^{+}_{\nu} W^{-}_{\nu})[/math][math]+g^{2}s_{w}c_{w}[A_{\mu}Z^{0}_{\nu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu}) -2A_{\mu}Z^{0}_{\mu}W^{+}_{\nu}W^{-}_{\nu}][/math][math]-g\alpha[H^3+H\phi^{0}\phi^{0}+2H\phi^{+}\phi^{-}][/math][math]-\frac{1}{8}g^{2}\alpha_{h}[H^4+(\phi^{0})^{4}+4(\phi^{+}\phi^{-})^{2}+4(\phi^{0})^{2}\phi^{+}\phi^{-}+4H^{2}\phi^{+}\phi^{-}+2(\phi^{0})^{2}H^{2}][/math][math]-gMW^{+}_{\mu}W^{-}_{\mu}H-\frac{1}{2}g\frac{M}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}H[/math][math]-\frac{1}{2}ig[W^{+}_{\mu}(\phi^{0}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{0}) -W^{-}_{\mu}(\phi^{0}\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}\phi^{0})][/math][math]+\frac{1}{2}g[W^{+}_{\mu}(H\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}H) -W^{-}_{\mu}(H\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}H)][/math][math]+\frac{1}{2}g\frac{1}{c_{w}}(Z^{0}_{\mu}(H\partial_{\mu}\phi^{0}-\phi^{0}\partial_{\mu}H) -ig\frac{s^{2}_{w}}{c_{w}}MZ^{0}_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})[/math][math]+igs_{w}MA_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})[/math][math]-ig\frac{1-2c^{2}_{w}}{2c_{w}}Z^{0}_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{- }\partial_{\mu}\phi^{+}) +igs_{w}A_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{+})[/math][math]-\frac{1}{4}g^{2}W^{+}_{\mu}W^{-}_{\mu}[H^{2}+(\phi^{0})^{2}+2\phi^{+}\phi^{-}][/math][math]-\frac{1}{4}g^{2}\frac{1}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}[H^{2}+(\phi^{0})^{2}+2(2s^{2}_{w}- 1)^{2}\phi^{+}\phi^{-}][/math][math]-\frac{1}{2}g^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-}_{\mu}\phi^{+}) -\frac{1}{2}ig^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})[/math][math]+\frac{1}{2}g^{2}s_{w}A_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-}_{\mu}\phi^{+}) +\frac{1}{2}ig^{2}s_{w}A_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})[/math][math]-g^{2}\frac{s_{w}}{c_{w}}(2c^{2}_{w}-1)Z^{0}_{\mu}A_{\mu}\phi^{+}\phi^{-}[/math][math]-g^{1}s^{2}_{w}A_{\mu}A_{\mu}\phi^{+}\phi^{-} -\bar{e}^{\lambda}(\gamma\partial+m^{\lambda}_{e})e^{\lambda} -\bar{\nu}^{\lambda}\gamma\partial\nu^{\lambda}[/math][math]-\bar{u}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{u})u^{\lambda}_{j} -\bar{d}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{d})d^{\lambda}_{j}[/math][math]+igs_{w}A_{\mu}[-(\bar{e}^{\lambda}\gamma^{\mu} e^{\lambda})+\frac{2}{3}(\bar{u}^{\lambda}_{j}\gamma^{\mu} u^{\lambda}_{j})-\frac{1}{3}(\bar{d}^{\lambda}_{j}\gamma^{\mu} d^{\lambda}_{j})][/math][math]+\frac{ig}{4c_{w}}Z^{0}_{\mu}[(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda})+(\bar{e}^{\lambda}\gamma^{\mu}(4s^{2}_{w}-1-\gamma^{5})e^{\lambda})+ (\bar{u}^{\lambda}_{j}\gamma^{\mu}(\frac{4}{3}s^{2}_{w}-1-\gamma^{5})u^{\lambda}_{j})+ (\bar{d}^{\lambda}_{j}\gamma^{\mu}(1-\frac{8}{3}s^{2}_{w}-\gamma^{5})d^{\lambda}_{j})][/math][math]+\frac{ig}{2\sqrt{2}}W^{+}_{\mu}[(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})e^{\lambda}) +(\bar{u}^{\lambda}_{j}\gamma^{\mu}(1+\gamma^{5})C_{\lambda\kappa}d^{\kappa}_{j})][/math][math]+\frac{ig}{2\sqrt{2}}W^{-}_{\mu}[(\bar{e}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda}) +(\bar{d}^{\kappa}_{j}C^{\dagger}_{\lambda\kappa}\gamma^{\mu}(1+\gamma^{5})u^{\lambda}_{j})][/math][math]+\frac{ig}{2\sqrt{2}}\frac{m^{\lambda}_{e}}{M} [-\phi^{+}(\bar{\nu}^{\lambda}(1-\gamma^{5})e^{\lambda}) +\phi^{-}(\bar{e}^{\lambda}(1+\gamma^{5})\nu^{\lambda})][/math][math]-\frac{g}{2}\frac{m^{\lambda}_{e}}{M}[H(\bar{e}^{\lambda}e^{\lambda}) +i\phi^{0}(\bar{e}^{\lambda}\gamma^{5}e^{\lambda})][/math][math]+\frac{ig}{2M\sqrt{2}}\phi^{+} [-m^{\kappa}_{d}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1-\gamma^{5})d^{\kappa}_{j}) +m^{\lambda}_{u}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1+\gamma^{5})d^{\kappa}_{j}][/math][math]+\frac{ig}{2M\sqrt{2}}\phi^{-} [m^{\lambda}_{d}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1+\gamma^{5})u^{\kappa}_{j}) -m^{\kappa}_{u}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1-\gamma^{5})u^{\kappa}_{j}][/math][math]-\frac{g}{2}\frac{m^{\lambda}_{u}}{M}H(\bar{u}^{\lambda}_{j}u^{\lambda}_{j}) -\frac{g}{2}\frac{m^{\lambda}_{d}}{M}H(\bar{d}^{\lambda}_{j}d^{\lambda}_{j}) +\frac{ig}{2}\frac{m^{\lambda}_{u}}{M}\phi^{0}(\bar{u}^{\lambda}_{j}\gamma^{5}u^{\lambda}_{j})[/math][math]-\frac{ig}{2}\frac{m^{\lambda}_{d}}{M}\phi^{0}(\bar{d}^{\lambda}_{j}\gamma^{5}d^{\lambda}_{j})[/math][math]+\bar{X}^{+}(\partial^{2}-M^{2})X^{+}+\bar{X}^{-}(\partial^{2}-M^{2})X^{-} +\bar{X}^{0}(\partial^{2}-\frac{M^{2}}{c^{2}_{w}})X^{0}+\bar{Y}\partial^{2}Y[/math][math]+igc_{w}W^{+}_{\mu}(\partial_{\mu}\bar{X}^{0}X^{-}-\partial_{\mu}\bar{X}^{+}X^{0}) +igs_{w}W^{+}_{\mu}(\partial_{\mu}\bar{Y}X^{-}-\partial_{\mu}\bar{X}^{+}Y)[/math][math]+igc_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}X^{0}-\partial_{\mu}\bar{X}^{0}X^{+}) +igs_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}Y-\partial_{\mu}\bar{Y}X^{+})[/math][math]+igc_{w}Z^{0}_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-}) +igs_{w}A_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-})[/math][math]-\frac{1}{2}gM[\bar{X}^{+}X^{+}H+\bar{X}^{-}X^{-}H+\frac{1}{c^{2}_{w}}\bar{X}^{0}X^{0}H][/math][math]+\frac{1-2c^{2}_{w}}{2c_{w}}igM[\bar{X}^{+}X^{0}\phi^{+}-\bar{X}^{-}X^{0}\phi^{-}] +\frac{1}{2c_{w}}igM[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}][/math][math]+igMs_{w}[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}] +\frac{1}{2}igM[\bar{X}^{+}X^{+}\phi^{0}-\bar{X}^{-}X^{-}\phi^{0}][/math]But that's not even the worst part! Trying to get a prediction out of this untidy jumble of Greek letters yields infinity in pretty much any nontrivial situation. So physicists had to introduce a trick called "renormalization" to remove the infinities. Renormalization doesn't have any particularly strong mathematical justification, but it gives the right results, so we kept using it.Then another problem came up: no one could explain why particles had mass. This was a… massive… problem. (Sorry.) In the 1960's, some clever physicists came up with a kludge called the Higgs mechanism that mostly filled the gap, but the ad hoc introduction of a new scalar field wasn't exactly satisfying. (All the same, I’ve included the Higgs mechanism in the Lagrangian above — it’s now an important, experimentally validated piece of the Standard Model.)In the end, we have the Standard Model: a patchwork quilt of quick fixes and unsightly workarounds that barely fits together. There are 17 fundamental particles (no one knows why) and somewhere around 18 arbitrarily adjustable parameters. Yet somehow this convoluted mishmash is the most accurate scientific theory of all time.For physicists, that’s a deeply unsettling state of affairs. We want to believe the universe isn’t really this ugly. We want a better theory to exist, a more beautiful theory that will unify quantum mechanics and general relativity in one simple and lovely set of equations. But so far, we haven’t been able to find anything convincing. A Nobel Prize awaits the first person who does.

Using the first principles definition of the derivative is a little tricky in this case but is certainly possible.[math]f(x) = \frac{1}{\sqrt{x}}[/math]Next we write what [math]f(x+h) [/math] is.[math]f(x+h) = \dfrac{1}{\sqrt{x+h}}[/math]Our formula for the derivative of [math]f(x)[/math] is the following.[math]f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}[/math]In the context of the problem…[math]f'(x) = \lim_{h \to 0} \dfrac{\left( \frac{1}{\sqrt{x+h}} \right) - \left( \frac{1}{\sqrt{x}} \right)}{h}[/math]Then multiply the first part in brackets by [math]\frac{\sqrt{x}}{\sqrt{x}}[/math] and the second part in brackets by [math]\frac{\sqrt{x + h}}{\sqrt{x + [/math][math]h[/math][math]}} [/math][math]f'(x) = \lim_{h \to 0} \dfrac{ \left( \frac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x}\sqrt{x+h}} \right)}{h}[/math]The next step is to remove that h in the denominator by multiplying by [math]\frac{1}{h}[/math][math]f'(x) = \lim_{h \to 0} \dfrac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}(\sqrt{x+h})}[/math]Then multiply this by [math]\frac{\sqrt{x} + \sqrt{x+h}}{\sqrt{x} + \sqrt{x+h}} [/math] to remove the surds from the numerator - this will hopefully simplify the problem.[math]f'(x) = \lim_{h \to 0} \frac{\sqrt{x} - \sqrt{x+h}}{h\sqrt{x}(\sqrt{x+h})} \cdot \frac{\sqrt{x} + \sqrt{x+h}}{\sqrt{x} + \sqrt{x+h}}[/math]You should know the expansion [math](a-b)(a+b) = a^2 - b^2[/math], so if you treat [math]\sqrt{x} - \sqrt{x+h}[/math] as [math](a-b)[/math] and [math]\sqrt{x} + \sqrt{x+h}[/math] as [math](a+b)[/math], then the numerator will be [math](\sqrt{x})^2 - (\sqrt{x +h})^2[/math] which is equal to [math]x-(x+h)[/math]TL;DR: use different of squares for the numerator.[math]f'(x) = \lim_{h \to 0} \dfrac{x - (x+h)}{hx\sqrt{x+h} + h\sqrt{x}(x+h)}[/math]Simplify the numerator and factor out [math]h[/math][math][/math] from the denominator.[math]f'(x)=\lim_{h \to 0} \dfrac{-h}{h(x \sqrt{x+h} + \sqrt{x}(x+h))}[/math]Cancel the [math]h[/math][math][/math] on the numerator and denominator.[math]f'(x)= \lim_{h \to 0} \dfrac{-1}{x \sqrt{x+h} + \sqrt{x}(x+h)}[/math]Substitute [math]h[/math][math] = 0[/math] (direct substitution now does not lead to an indeterminate form).[math]f'(x) = \dfrac{-1}{x \sqrt{x} + [/math][math]x[/math][math] \sqrt{x}}[/math]Collect like terms.[math]f'(x) = \dfrac{-1}{2x \sqrt{x}}[/math]There are a few ways you can express this by saying that.[math]f'(x) = \dfrac{-1}{2x^{\frac{3}{2}}}[/math]Or even.[math]f'(x) = \dfrac{-x^{\frac{-3}{2}}}{2}[/math]All ways of expressing the derivative are correct though.Also in the 6th image I removed the [math]h[/math][math][/math] from the denominator by multiplying by [math]\frac{1}{h}[/math] but you don’t have to do it that way to get the right answer. You can simply continue to do the difference of squares regardless and remove it at a later step.Thanks and hope I could help!Edit: As pointed out by Aaron Dunbrack there was a mistake in the working out which has since been corrected. If anyone finds any more mistakes please do let me know

What makes a quadratic equation [math]ax^2 + bx + c = 0[/math] hard to solve? It’s that pesky degree-1 term [math]bx[/math]. Without it, we could easily solve[math]ax^2 + c = 0 \iff [/math][math]x[/math][math] = \pm \sqrt{-\tfrac ca}[/math].But how can we get rid of the degree-1 term? We can try a substitution [math]x[/math][math]' = [/math][math]x[/math][math] - [/math][math]h[/math][math][/math]:[math]ax^2 + bx + c = 0[/math][math]\iff a(x' + h)^2 + b(x' + [/math][math]h[/math][math]) + c = 0[/math][math]\iff ax'^2 + (2ah + b)x' + (ah^2 + bh + c) = 0[/math]and now we see that the degree-1 term will disappear as long as[math]2ah + [/math][math]b[/math][math] = 0 \iff [/math][math]h[/math][math] = -\frac{b}{2a}[/math].The general idea of making a substitution with a degree of freedom (here the unknown constant [math]h[/math][math][/math]), and then imposing an arbitrary constraint to simplify the resulting equation, is very powerful—powerful enough, in fact, to solve the cubic equation [math]ax^3 + bx^2 + cx + d = 0[/math] too.First we use the same substitution [math]x[/math][math]' = [/math][math]x[/math][math] - [/math][math]h[/math][math][/math] in order to eliminate the degree-2 term. While we’re at it, we’ll divide everything by [math]a[/math] so that the equation is monic. Now it looks like [math]x'^3 + px' + q = 0[/math]. Let’s try another substitution [math]x[/math][math]' = u + v[/math], which turns our equation into[math]u^3 + 3u^2v + 3uv^2 + v^3 + pu + pv + q = 0[/math][math]\iff u^3 + v^3 + (3uv + p)(u + v) + q = 0[/math].We have a degree of freedom in the way we choose [math]u[/math] and [math]v[/math], so let’s use it to cancel out the middle term by imposing [math]3uv + p = 0[/math].We have just enough room to solve for [math]u[/math] and [math]v[/math] in terms of [math]p[/math] and [math]q[/math]:[math]u^3v^3 = -\frac{p^3}{27}[/math][math]u^3 + v^3 = -q[/math]so [math]z_1 = u^3[/math] and [math]z_2 = v^3[/math] are the roots of the quadratic equation[math]z^2 + qz - \frac{p^3}{27} = 0[/math].We can take the three complex cube roots of [math]z_1[/math] and [math]z_2[/math] to find three solutions for [math]x[/math][math]' = u + v[/math].An even more complicated instance of the same idea is used to reduce a quartic equation to a cubic one.