Derivatives and Limits

Derivatives

• Derivatives is finding the instantaneous rate of change of a function at a particular point.

  • Imagine a curve; at a given point, you draw a tangent. The slope of this tangent at that point gives the derivative value.
  • Slope is change in y ($\Delta y$) divided by change in x ($\Delta x$).

Limits

• Limits help when you can't directly calculate a function's value at a point.

  • Instead, you examine the function's behavior as you approach that point from the left and right.
  • If $f(x) = x + 1$, you can't directly solve for $x = 1$, you can approach $x = 1$ from the left (e.g., 0.9, 0.99, 0.999) and from the right (e.g., 1.1, 1.01, 1.001).
  • As $x$ gets closer to 1, $y$ gets closer to 2.

• Let's say $f(x) = \frac{x^2 - 4}{x - 2}$. Here, $x$ cannot be 2 because the denominator would be zero (undefined).

  • However, you can simplify the function to $f(x) = x + 2$, but remember that the original function still restricts $x$ from being 2.
  • As $x$ approaches 2, $f(x)$ approaches 4.
  • Limits define constraint values when you can't use one of the direct values.

Left Hand Limit (LHL) and Right Hand Limit (RHL)

• When approaching from the left, it's the Left Hand Limit (LHL).
• When approaching from the right, it's the Right Hand Limit (RHL).
• For example, if $f(x) = x + 2$, then:
  • $\lim_{x \to 2^-} f(x) = 4$ (LHL)
  • $\lim_{x \to 2^+} f(x) = 4$ (RHL)
• Limits value indicate how close you can get to a certain value.

Derivatives Formula

• The derivative of a function is written as $f'(x)$.

• First Principle: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

  • This formula calculates the slope of the tangent line at a point on the curve.
  • It's derived from the basic slope formula: $\frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
  • Here, we're taking the initial point as $x$ and the final point as $x + h$, where $h$ is a very small quantity tending to zero.

• $f'(x)$ is the derivative of $f(x)$, and $f(x)$ is often denoted as $y$.

  • If $y = 3x^3$, then $f'(x)$ is the derivative of $y$, written as $\frac{dy}{dx}$ (change in $y$ with respect to $x$).

Derivative of Sine

• $\frac{d}{dx} \sin(x) = \cos(x)$
• Proof using the first principle:
  • If $f(x) = \sin(x)$, then $f(x + h) = \sin(x + h)$.
  • Using the formula, $f'(x) = \lim_{h \to 0} \frac{\sin(x + h) - \sin(x)}{h}$
  • Applying the trigonometric identity, $f'(x) = \lim_{h \to 0} \frac{\sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)}{h}$
  • Rearranging, $f'(x) = \lim_{h \to 0} \frac{\sin(x)(\cos(h) - 1) + \cos(x)\sin(h)}{h}$
  • Separating the limits, $f'(x) = \lim_{h \to 0} \sin(x) \frac{(\cos(h) - 1)}{h} + \cos(x) \frac{\sin(h)}{h}$
  • Using standard limits, $f'(x) = \sin(x) \cdot 0 + \cos(x) \cdot 1 = \cos(x)$

Standard Limits

• $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$
• $\lim_{x \to 0} \frac{\tan(x)}{x} = 1$
• $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$
• Using Trigonometry: $1 - \cos(\theta) = 2\sin^2(\frac{\theta}{2})$

L'Hôpital's Rule

• If a limit is in the indeterminate form $\frac{0}{0}$, you can differentiate the numerator and denominator separately.
• For example, to prove $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$:
  • Differentiate the numerator and denominator: $\lim_{x \to 0} \frac{\cos(x)}{1}$
  • Now, $\cos(0) = 1$, so the limit is 1.
• To prove $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$:
  • Differentiate: $\lim_{x \to 0} \frac{\sin(x)}{1}$
  • Since $\sin(0) = 0$, the limit is 0.

Other Derivatives

• $\frac{d}{dx} \cos(x) = -\sin(x)$

  • Using the first principle:
  • If $f(x) = \cos(x)$, then $f(x+h) = \cos(x+h)$
  • $f'(x) = \lim_{h \to 0} \frac{\cos(x+h) - \cos(x)}{h}$
  • $f'(x) = \lim_{h \to 0} \frac{\cos(x)\cos(h) - \sin(x)\sin(h) - \cos(x)}{h}$
  • $f'(x) = \lim_{h \to 0} \cos(x) \frac{\cos(h) - 1}{h} - \sin(x) \frac{\sin(h)}{h}$
  • Then: $f'(x) = \cos(x) * 0 - \sin(x) * 1 = -\sin(x)$

• $\frac{d}{dx} \tan(x) = \sec^2(x)$

  • You can use the quotient rule with $\tan(x) = \frac{\sin(x)}{\cos(x)}$ or use the identity $\tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}$

• $\frac{d}{dx} \cot(x) = -\csc^2(x)$

• $\frac{d}{dx} \sec(x) = \sec(x)\tan(x)$

• $\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)$

Power Rule

• $\frac{d}{dx} x^n = nx^{n-1}$
  • Example: If $f(x) = x^5$, then $f'(x) = 5x^4$.
  • If $f(x) = x^{100}$, then $f'(x) = 100x^{99}$.

Exponential Rules

• $\frac{d}{dx} e^x = e^x$
• $\frac{d}{dx} a^x = a^x \ln(a)$

Logarithmic Equations

• $\frac{d}{dx} \log(x) = \frac{1}{x}$
  • Note: If no base is specified, then $\log(x)$ refers to $\log_{10}(x)$. When integrating $\frac{1}{x}$, the result will be $\ln(x)$, assuming the base is $e$.
• $\frac{d}{dx} \log<em>a(x) = \frac{1}{x} \log</em>e(a)$

Algebra of Derivatives

• If you have a sum or difference of functions, you can differentiate them separately.
• If you have a constant multiplied by a function, you can pull the constant outside of the derivative.
• Example: If $f(x) = 5x^{50}$, then $f'(x) = 5 * 50x^{49} = 250x^{49}$.

Product Rule:

• If you have two functions multiplied together, you can't just differentiate them separately; you must use the product rule.
• $(f(x) * g(x))' = f'(x) * g(x) + f(x) * g'(x)$
  • You keep one function constant and differentiate the other function, then add the derivative of the first function and multiplying the second.
• Example: $\frac{d}{dx} (x \sin(x)) = x \cos(x) + \sin(x)$
• For three variables: d/dx(x * e^x * sin x ) = x * e^x * cos x + x * sin x * e^x + sin x * e^x * 1

Quotient Rule:

• If you have a function in the form of $\frac{u}{v}$, then the derivative is:$\Big(\frac{u}{v}\Big)' = \frac{v * u' - u * v'}{v^2}$. You have to keep the denominator times derivative of the numerator, times the numerator times derivative of the denominator, all divided by the denominator squared.
• For derivatives of division form, the square of the bottom function is at the denominator of the answer after differentiating.

Chain Rule:

• Chain Rule is applied when you have a function within another function (ex: $f(g(x))$).
• First, we differentiate the outermost function and then the innermost function, giving multiplication sign in between them. For example, $\frac{d}{dx} sin(x^2) = cos(x^2) * 2x$
• d/dx is equivalent to saying ddy/dx. This can also be expressed as ddy/du times ddu/dx.

Logarithms Reminders

• Logarithm (log) is a common logarithm. Generally, you take the base either 10 or something. So when you are writing log x, just log x, you understand them by writing log x base 10. This is called common logarithm.
• ln is known as Natural logarithm. This is known as ln x, ln x. So the difference is for the bass.
• Remember that $$", Ln a is log a base e"