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Math Analysis 1

Derivatives: definition, table and differentiation rules

Difference quotient and tangent line, corners and cusps, derivatives of elementary functions, differentiation rules with examples, the differential and common mistakes.

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The derivative measures how fast a function changes. Geometrically it is the slope of the tangent line to the graph, physically it is a rate: current is the derivative of charge, the voltage across an inductor is proportional to the derivative of the current.

Definition

f is differentiable at x_0 if the limit of the difference quotient exists and is finite:

    \[ f'(x_0) = \lim_{h\to 0}\frac{f(x_0 + h) - f(x_0)}{h}. \]

The quotient is the slope of the line through two points of the graph; in the limit it becomes the slope of the tangent, whose equation is y = f(x_0) + f'(x_0)(x - x_0).

Differentiable implies continuous, but not the other way round. |x| is continuous at 0 but has right derivative +1 and left derivative -1: a corner. \sqrt[3]{x} at 0 has a vertical tangent (infinite derivative), \sqrt{|x|} has a cusp.

Derivatives of elementary functions

f(x)f'(x)f(x)f'(x)
x^\alpha\alpha x^{\alpha-1}\tan x1 + \tan^2 x = \frac{1}{\cos^2 x}
e^xe^x\arcsin x\frac{1}{\sqrt{1-x^2}}
a^xa^x \ln a\arccos x-\frac{1}{\sqrt{1-x^2}}
\ln x\frac1x\arctan x\frac{1}{1+x^2}
\sin x\cos x\ln|x|\frac1x
\cos x-\sin x\sqrt{x}\frac{1}{2\sqrt x}

Each one follows from the definition and a standard limit. For instance \frac{\sin(x+h) - \sin x}{h} = \sin x\,\frac{\cos h - 1}{h} + \cos x\,\frac{\sin h}{h} \to \cos x.

Differentiation rules

  • Linearity: (af + bg)' = af' + bg'.
  • Product: (fg)' = f'g + fg' (proof in the in-depth article).
  • Quotient: \left(\frac fg\right)' = \frac{f'g - fg'}{g^2}.
  • Composite function (chain rule): \big(f(g(x))\big)' = f'(g(x))\cdot g'(x).
  • Inverse function: if y = f(x), (f^{-1})'(y) = \frac{1}{f'(x)}. This gives (\ln x)' = \frac{1}{e^{\ln x}} = \frac1x.

Examples.

  • \big(\sin(x^2)\big)' = 2x\cos(x^2);
  • \big(e^{-3x}\cos x\big)' = -3e^{-3x}\cos x - e^{-3x}\sin x;
  • \big(\ln(1 + x^2)\big)' = \frac{2x}{1+x^2};
  • \big(x^x\big)' = \big(e^{x\ln x}\big)' = x^x(\ln x + 1): when the variable is both in the base and in the exponent, switch to the exponential.

Higher derivatives and the differential

Differentiating repeatedly gives f'', f''', f^{(n)}. The second derivative measures how the slope changes (in physics it is acceleration) and decides the concavity of the graph. The differential df = f'(x_0)\,dx is the change along the tangent: for small increments f(x_0 + h) \approx f(x_0) + f'(x_0)h. It is the linear approximation used to estimate errors: if R = V/I and the voltage has a 1% relative error, R also gets about 1% error from V.

Common mistakes

  • Forgetting the inner derivative in the chain rule: (\cos 3x)' = -3\sin 3x, not -\sin 3x.
  • Using the power rule for exponentials: (2^x)' = 2^x\ln 2, not x\,2^{x-1}.
  • Differentiating |x| as if it were x: the derivative is \operatorname{sgn} x for x \ne 0 and does not exist at 0.

In circuits

The constitutive relations of components are derivatives: i = C\frac{dv}{dt} for a capacitor, v = L\frac{di}{dt} for an inductor. With a voltage v(t) = V_m\sin(\omega t) across a capacitor, the current is C\omega V_m\cos(\omega t): leading by 90° and proportional to frequency. That is why a capacitor “blocks” DC and lets high frequencies through.

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