Differentiation Techniques

• 1. Intuitive Limits
• 2. Limit of Exponential Functions
• 3. Inverse Trigonometric Functions
• 4. Applying the Mean Value Theorem to Limits
• 5. Extrema of Rational Functions
• 6. Derivative of an Absolute Value Function
• 7. Derivative & Errors
• 8. Asymptotes
• 9. Curvature
• 10. Gradient
• 11. Directional Derivative
• 12. Derivative of an Implicit Function
• 13. Bivariate Extreme Value
• 14. Extrema of an Implicit Function
• 15. Lagrange Multiplier

1. Intuitive Limits

The limit of a function as $x \to 0$, this function can be replaced so that calculating the limit is much simpler. The following replacements can be considered.

• $\sin x \implies x$,
  • $\sin kx \implies kx$, $\; \sin^n x \implies x^n$
• $\tan x \implies x$,
  • $\tan kx \implies kx$, $\; \tan^n x \implies x^n$
• $\sin^{-1} x \implies x$,
  • $\sin^{-1} kx \implies kx$, $\; (\sin^{-1} kx)^n \implies (kx)^n$
• $\tan^{-1} x \implies x$,
  • $\tan^{-1} kx \implies kx$, $\; (\tan^{-1} kx)^n \implies (kx)^n$
• $x^3 + x \implies x$.
  • $x^5 - x^3 + x \implies x$, so in this case addition/subtraction does not matter.

When $x$ is large enough, the following comparison is also useful. $$\begin{aligned} \vert \sin x \vert < \ln x < x^n < 2^x < x! < x^x \end{aligned}$$

2. Limit of Exponential Functions

There are a few ways to find the limits of the form of $0^0$, $\infty^0$, or $1^\infty$.

• Given $y = f(x)^{g(x)}$, taking a logarithm of both sides of the function may be helpful. In other words, the limit of $y$ can be found after calculating the limit of $\ln y = g(x) \ln f(x)$.

• The property $a^b = e^{b \ln a}$ may be used.

$$\begin{aligned} \begin{cases} \; \displaystyle \lim_{x \to a} f(x)^{g(x)} = \displaystyle \lim_{x \to a} e^{g(x) \ln {f(x)}} \\\\ \; \displaystyle \lim_{x \to a} f(x)^{\frac{1}{g(x)}} = \displaystyle \lim_{x \to a} e^{\frac{\ln {f(x)}}{g(x)}} \end{cases} \end{aligned}$$

• Especially, in the case of $1^\infty$, the property $a^b = e^{b (a-1)}$ may be used as well. Assuming that $f(a) = 1$ and $g(a) = 0$,

$$\begin{aligned} \lim_{x \to a} \cfrac{\ln {f(x)}}{g(x)} = \lim_{x \to a} \cfrac{\cfrac{f'(x)}{f(x)}}{g'(x)} &= \lim_{x \to a} \cfrac{f'(x)}{g'(x)} = \lim_{x \to a} \cfrac{\cfrac{f(x) - f(a)}{x - a}}{\cfrac{g(x) - g(a)}{x - a}} = \lim_{x \to a} \cfrac{f(x) - 1}{g(x)} \\\\ \implies \lim_{x \to a} f(x)^{\frac{1}{g(x)}} &= \lim_{x \to a} e^{\frac{\ln {f(x)}}{g(x)}} = \lim_{x \to a} e^{\frac{f(x) - 1}{g(x)}} \end{aligned}$$

• The following results are also useful.

$$\begin{aligned} \begin{cases} \; \displaystyle \lim_{x \to 0} (kx)^{hx} = 1 \\\\ \; \displaystyle \lim_{x \to \infty} (kx)^{\frac{h}{x}} = 1 \end{cases}, \qquad \begin{cases} \; \displaystyle \lim_{x \to 0} \left(\cfrac{k}{x}\right)^{hx} = 1 \\\\ \; \displaystyle \lim_{x \to \infty} \left(\cfrac{k}{x}\right)^{\frac{h}{x}} = 1 \end{cases} \end{aligned}$$

3. Inverse Trigonometric Functions

Derivatives of inverse trigonometric functions are as follows.

• $(\sin^{-1} x)' = \cfrac{1}{\sqrt{1 - x^2}} \quad \left( -\cfrac{\pi}{2} \leq \sin^{-1} x \leq \cfrac{\pi}{2} \right) \;$,
• $(\cos^{-1} x)' = \cfrac{-1}{\sqrt{1 - x^2}} \quad \left( 0 \leq \cos^{-1} x \leq \pi \right) \;$,
• $(\tan^{-1} x)' = \cfrac{1}{1 + x^2} \quad \left( -\cfrac{\pi}{2} < \tan^{-1} x < \cfrac{\pi}{2} \right) \;$,
• $(\cot^{-1} x)' = -\cfrac{1}{1 + x^2} \quad \left( 0 < \cot^{-1} x < \pi \right) \;$,
• $(\sec^{-1} x)' = \cfrac{1}{\vert x \vert \sqrt{x^2 - 1}} \quad \left( 0 \leq \sec^{-1} x \leq \pi, \; \sec^{-1} x \ne \cfrac{\pi}{2} \right) \;$,
• $(\csc^{-1} x)' = -\cfrac{1}{\vert x \vert \sqrt{x^2 - 1}} \quad \left( -\cfrac{\pi}{2} \leq \csc^{-1} x \leq \cfrac{\pi}{2}, \; \csc^{-1} x \ne 0 \right) \;$.

In addition, the following are important identities of inverse trigonometric functions.

• $\sin^{-1}x + \cos^{-1}x = \cfrac{\pi}{2} \;$,
• $\sin^{-1}(-x) = -\sin^{-1}x \;$,
• $\cos^{-1}(-x) = \pi - \cos^{-1}x \;$,
• $\csc^{-1} x = \sin^{-1}\left( \cfrac{1}{x} \right) \;$.

4. Applying the Mean Value Theorem to Limits

If the function $f(x)$ is continuous on the closed interval $\left[g(x), h(x)\right]$ and differentiable on the open interval $\left(g(x), h(x)\right)$, and $\lim\limits_{x \to a} g(x) = \lim\limits_{x \to a} h(x) = k$, then $$\begin{aligned} \lim_{x \to a} \frac{f(h(x)) - f(g(x))}{h(x) - g(x)} = f'(k). \end{aligned}$$

By the Mean Value Theorem, there exists at least one $c$ in the open interval $\left(g(x), h(x)\right)$ such that $$\begin{aligned} \frac{f(h(x)) - f(g(x))}{h(x) - g(x)} = f'(c). \end{aligned}$$

As the open interval $\left(g(x), h(x)\right)$ shrinks down to the point $k$ in the limit, $c$ must still lie inside this interval, so $c$ converges to $k$. In other words, the average rate of change over the open interval $\left(g(x), h(x)\right)$ converges to the instantaneous rate of change at $k$. Therefore, $$\begin{aligned} \lim_{x \to a} \frac{f(h(x)) - f(g(x))}{h(x) - g(x)} = \lim_{x \to a} f'(c) = \lim_{c \to k} f'(c) = f'(k). \end{aligned}$$

5. Extrema of Rational Functions

For $g(x)/f(x)$, let $\alpha$ be a root of the equation $g(x)/f(x) = g'(x)/f'(x)$. Then the extreme value of $g(x)/f(x)$ is $$\begin{aligned} \cfrac{g'(\alpha)}{f'(\alpha)} \end{aligned}$$

Let $h(x) = g(x) / f(x)$. Then $f(x) h(x) = g(x)$. Differentiating both sides, $f'(x) h(x) + f(x) h'(x) = g'(x)$. $$\begin{aligned} h'(x) = \cfrac{g'(x) - f'(x) h(x)}{f(x)} \end{aligned}$$

If $h(x)$ has an extremum at $x = \alpha$, then $h'(\alpha) = 0$, so $$\begin{aligned} h'(\alpha) = \cfrac{g'(\alpha) - f'(\alpha) h(\alpha)}{f(\alpha)} = 0 &\implies g'(\alpha) = f'(\alpha) h(\alpha) \\ &\implies h(\alpha) = \cfrac{g'(\alpha)}{f'(\alpha)} \qquad \left( f'(\alpha) \neq 0 \right) \end{aligned}$$

This method gives the extreme value, but whether it is a local maximum or a local minimum must be determined by another method. If the shape of the graph is known, local maxima and minima can be identified easily.

6. Derivative of an Absolute Value Function

Usually, although this can be done by range, the following formula is noteworthy. $$\begin{aligned} (\vert f(x) \vert)' = \cfrac{f(x) f(x)'}{\vert f(x) \vert} \end{aligned}$$

7. Derivative & Errors

By definition, the derivative of $f(x)$ is as follows. $$\begin{aligned} \lim_{\Delta x \to 0} \cfrac{\Delta y}{\Delta x} = f'(x) \end{aligned}$$

Since the average rate of change is not the same as the instantaneous rate of change in the view of the limit, the difference can be set as follows. $$\begin{aligned} \cfrac{\Delta y}{\Delta x} = f'(x) + \epsilon \end{aligned}$$

Note that $\epsilon$ is smaller when $\Delta x$ gets smaller. It implies that the product of them is an infinitesimal number and it can be ignored. Therefore, the approximation of $\Delta y$ is $f'(x) \Delta x$. The limit of this approximation is called the derivative of $y$ and is written as $$\begin{aligned} dy = df(x) = f'(x) dx \end{aligned}$$

Another point of view of $dy$ is that it is the error of $y$. Moreover, $dy/y$ is called the relative error. Meanwhile, this concept leads to the approximation formula about $f(x + \Delta x)$. $$\begin{aligned} f(x + \Delta x) \approx f(x) + f'(x) \Delta x \end{aligned}$$

8. Asymptotes

Asymptotes make it easier to draw a graph of a function. Vertical asymptotes can be found when a function goes to infinity as $x$ approaches a value. Also, horizontal asymptotes can be found when a function goes to a value as $x$ approaches $\pm \infty$. Other than these, there exist oblique asymptotes, which can be found in the following steps.

• Divide both sides of the function by the highest order term and find a slope $m$ of the asymptote by evaluating the limits at infinity.
• Set $y = mx + n$ from the found $m$ and substitute this equation to the function. Again, divide both sides by the highest order term and find $n$ by evaluating the limits at infinity.

For example, the oblique asymptotes of $y^3 = (x - 1)(x - 3)^2$ can be found as follows. $$\begin{aligned} &\lim_{x \to \infty} \left( \cfrac{y}{x} \right)^3 = \lim_{x \to \infty} \cfrac{(x - 1)(x - 3)^2}{x^3} = 1 \\\\ &\implies y = x + n \\ &\implies (x + n)^3 = (x - 1)(x - 3)^2 = x^3 - 7x^2 + 15x - 9 \\ &\implies 3nx^2 + 3n^2x + n^3 = -7x^2 + 15x - 9 \\ &\implies \lim_{x \to \infty} \cfrac{3nx^2 + 3n^2x + n^3}{x^2} = \lim_{x \to \infty} \cfrac{-7x^2 + 15x - 9}{x^2} \\ &\implies 3n = -7 \\ &\implies y = x - \frac{7}{3} \end{aligned}$$

9. Curvature

Let $y = f(x)$ be a twice-differentiable curve on a plane. Now, consider $P(x, y)$ and $Q(x + \Delta x, y + \Delta y)$. When $\alpha$ is the angle between the tangent line on $P$ and $x$-axis and $\alpha + \Delta \alpha$ is the angle between the tangent line on $Q$ and $x$-axis, the tangent line is rotated by $\Delta \alpha$ moving the arc length $\Delta s$ on the curve. The mean curvature denotes the ratio of $\Delta \alpha$ to $\Delta s$. Based on this concept, curvature $K$ of $f(x)$ is computed as follows. $$\begin{aligned} K = \cfrac{\vert y'' \vert}{(1 + (y')^2)^{\frac{3}{2}}} \end{aligned}$$

If this curve is a parametric function, $K$ is $$\begin{aligned} \begin{cases} \; x = f(t) \\ \; y = g(t) \end{cases} \implies K = \cfrac{\vert x' \cdot y'' - x'' \cdot y' \vert}{((x')^2 + (y')^2)^{\frac{3}{2}}} \end{aligned}$$

Meanwhile, the radius of curvature $R$ can be calculated from $K$. $$\begin{aligned} K = \cfrac{1}{R} \end{aligned}$$

Furthermore, the center of curvature $(X, Y)$ on $P(x, y)$ can be calculated as follows. $$\begin{aligned} X = x - \cfrac{y' (1 + (y')^2)}{y''}, \quad Y = y + \cfrac{1 + (y')^2}{y''} \end{aligned}$$

10. Gradient

The gradient has different definitions depending on whether the function is explicit or implicit. $$\begin{aligned} \begin{cases} \; \text{Explicit Function } z = f(x, y) \implies \nabla f = \cfrac{\partial f}{\partial x} i + \cfrac{\partial f}{\partial y} j \\\\ \; \text{Implicit Function } f(x, y, z) = 0 \implies \nabla f = \cfrac{\partial f}{\partial x} i + \cfrac{\partial f}{\partial y} j + \cfrac{\partial f}{\partial z} k \end{cases} \end{aligned}$$

The gradient of an explicit function means the tangent vector at the point, while the gradient of an implicit function means the normal vector at the point. The details about this can be found in this note. Therefore, the normal vector from an explicit function can be obtained after reforming to an implicit function as follows. $$\begin{aligned} z - f(x, y) = 0 \implies \left( -\cfrac{\partial f}{\partial x}, \; -\cfrac{\partial f}{\partial y}, \; 1 \right) \end{aligned}$$

11. Directional Derivative

The directional derivative of a multivariable differentiable function $f$ along a given vector $u$ at a given point $x$ represents the slope of the tangent at $x$ with the direction $u$. If $u$ is parallel with the coordinate axes, the directional derivative is the partial derivative. Accordingly, the directional derivative can be viewed as an orthogonal projection of the partial derivative on $u$. Consider $P(a, b)$ on the function $f(x, y)$. The directional derivative along the normalized direction $u$ at the point $P$ is $$\begin{aligned} D_u f(P) &= \lim_{h \to 0} \cfrac{f(a + hu_x, b + hu_y) - f(a, b)}{h} = \nabla f(P) \cdot u \\\\ &= \left( \cfrac{\partial f}{\partial x} i + \cfrac{\partial f}{\partial y} j \right) \cdot (u_x i + u_y j) = \cfrac{\partial f}{\partial x}u_x + \cfrac{\partial f}{\partial y}u_y \end{aligned}$$

Therefore, the maximum of $D_u f(P)$ is $\Vert \nabla f(P) \Vert$ and the minimum is $-\Vert \nabla f(P) \Vert$.

12. Derivative of an Implicit Function

The general formula for the derivative of an implicit function is as follows. $$\begin{aligned} \begin{cases} \; f(x, y) = 0 \implies \cfrac{dy}{dx} = -\cfrac{f_x}{f_y} \\\\ \; f(x, y, z) = 0 \implies \cfrac{\partial z}{\partial x} = -\cfrac{f_x}{f_z}, \quad \cfrac{\partial z}{\partial y} = -\cfrac{f_y}{f_z} \end{cases} \end{aligned}$$

13. Bivariate Extreme Value

To find extreme values of a function $f(x, y)$, the point $(x, y)$ such that $f_x = 0$ and $f_y = 0$ should be calculated. Let that point be $(a, b)$. If $D$ is defined as follows, $(a, b)$ is the extreme value when $D>0$. $$\begin{aligned} D &= f_{xx}(a, b) f_{yy}(a, b) - (f_{xy}(a, b))^2 \\\\ D > 0 &\implies \begin{cases} \; f_{xx}(a, b) > 0 & \text{local minimum} \\\\ \; f_{xx}(a, b) < 0 & \text{local maximum} \end{cases} \end{aligned}$$

If $D<0$, there are no extreme values. Besides, if $D = 0$, it is inconclusive.

14. Extrema of an Implicit Function

The extrema of $y$ of an implicit function $f(x, y) = 0$ can be calculated from $f_x = 0$ and $f(x, y) = 0$. Let such point be $(a, b)$. If $f_y \ne 0$, $$\begin{aligned} \begin{cases} \; -\cfrac{f_{xx}(a, b)}{f_y(a,b)} > 0 & \text{local minimum } b \\\\ \; -\cfrac{f_{xx}(a, b)}{f_y(a,b)} < 0 & \text{local maximum } b \end{cases} \end{aligned}$$

It is inconclusive if $f_{xx}(a, b) = 0$.

15. Lagrange Multiplier

When finding the maximum and minimum of $f(x)$, this problem becomes often easy if an unknown variable can be represented using other unknown variables under constraints. Even so, however, the problem can sometimes get complex, and the better method is needed for this case. First, if a constraint $g(x, y, z) = 0$ is the only one, $$\begin{aligned} \begin{cases} \; \nabla f = \lambda \nabla g \\\\ \; g(x, y, z) = 0 \end{cases} \end{aligned}$$

the maximum or minimum can be found by calculating $\lambda$ and $(x_0, y_0, z_0)$ satisfying the above conditions. Here $\lambda$ is called the Lagrange multiplier. If another constraint $h(x, y, z) = 0$ is added, $$\begin{aligned} \begin{cases} \; \nabla f = \lambda \nabla g + \mu \nabla h \\\\ \; g(x, y, z) = 0 \\\\ \; h(x, y, z) = 0 \end{cases} \end{aligned}$$

the maximum or minimum can be found by calculating $\lambda$, $\mu$, and $(x_0, y_0, z_0)$ satisfying the above conditions. For example, the intersection of a cylinder $y^2 + z^2 = 4$ and a plane $x + y + z = 1$ forms an ellipse. The Lagrange multiplier makes it simple to find the maximum and minimum of $f = x + 2y$ for a point on this ellipse as follows. $$\begin{aligned} f &= x + 2y, \quad \begin{cases} \; g = x + y + z - 1 = 0 \\ \; h = y^2 + z^2 - 4 = 0 \end{cases} \\\\ &\implies \nabla f = (1, 2, 0), \quad \begin{cases} \; \nabla g = (1, 1, 1) \\ \; \nabla h = (0, 2y, 2z) \end{cases} \\\\ &\implies \nabla f = \lambda \nabla g + \mu \nabla h \iff (1, 2, 0) = \lambda(1, 1, 1) + \mu(0, 2y, 2z) \\ &\implies \lambda = 1, \quad \lambda + 2y\mu = 2, \quad \lambda + 2z\mu = 0 \\ &\implies \lambda = 1, \quad y = \cfrac{1}{2\mu}, \quad z = -\cfrac{1}{2\mu} \\ &\implies h = y^2 + z^2 - 4 = \cfrac{1}{4\mu^2} + \cfrac{1}{4\mu^2} - 4 = 0 \\ &\implies \mu = \pm \cfrac{1}{2\sqrt{2}}, \quad y = \pm 2\sqrt{2}, \quad z = \mp 2\sqrt{2} \\ &\implies g = x + y + z - 1 = x - 1 = 0 \\ &\implies x = 1 \end{aligned}$$

Therefore, the maximum is $f(1, \sqrt{2}, -\sqrt{2}) = 1 + 2\sqrt{2}$, and the minimum is $f(1, -\sqrt{2}, \sqrt{2}) = 1 - 2\sqrt{2}$.