Comprehensive Study Guide for Differential Calculus: Fundamentals, Rules, and Applications

The derivative of a function represents the instantaneous rate of change of the function with respect to its variable. Geometrically, it represents the slope of the tangent line to the curve at a specific point.
The derivative is formally defined using the limit of the difference quotient: * $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
The difference quotient is derived from the slope formula for a secant line passing through points $(x, f(x))$ and $(x+h, f(x+h))$ : * $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{f(x+h) - f(x)}{(x+h) - x} = \frac{f(x+h) - f(x)}{h}$
As $h$ approaches zero, the secant line becomes the tangent line at the point $x$ .
The derivative of a constant function $f(x) = c$ is always zero: * $\frac{d}{dx}(c) = 0$
The derivative of a linear function $f(x) = mx + b$ is simply the slope $m$ : * $\frac{d}{dx}(mx + b) = m$

The Power Rule is a primary shortcut used to differentiate power functions of the form $f(x) = x^n$ , where $n$ is any real number: * $\frac{d}{dx}(x^n) = n \times x^{n-1}$
Application to Radicals: To differentiate roots, they must first be rewritten as fractional exponents: * $\sqrt{x} = x^{\frac{1}{2}} \rightarrow \frac{d}{dx}(x^{\frac{1}{2}}) = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$ * $\sqrt[n]{x^m} = x^{\frac{m}{n}}$
Application to Reciprocals: To differentiate variables in the denominator, use negative exponents: * $\frac{1}{x^n} = x^{-n} \rightarrow \frac{d}{dx}(x^{-n}) = -n \times x^{-n-1}$
Constant Multiple Rule: If a constant is multiplied by a function, the derivative is the constant times the derivative of the function: * $\frac{d}{dx}(c \times f(x)) = c \times f'(x)$
Sum and Difference Rule: The derivative of a sum or difference is the sum or difference of the derivatives: * $\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x)$

The Product Rule is used when a function is the product of two differentiable sub-functions, $f(x) = u(x) \times v(x)$ .
The formula is defined as: * $\frac{d}{dx}[u(x) \times v(x)] = u'(x) \times v(x) + u(x) \times v'(x)$
In verbal terms: "The derivative of the first times the second, plus the first times the derivative of the second."
Example calculation for $f(x) = x^2 \sin(x)$ : * Let $u = x^2$ and $v = \sin(x)$ . * $u' = 2x$ and $v' = \cos(x)$ . * $f'(x) = (2x)(\sin(x)) + (x^2)(\cos(x))$

The Quotient Rule is applied when differentiating a function in the form of a fraction, $f(x) = \frac{u(x)}{v(x)}$ , where $v(x) \neq 0$ .
The formula is defined as: * $\frac{d}{dx}\left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$
A common mnemonic is "Low D-High minus High D-Low, over Low-Low."
Order matters in the numerator because of the subtraction sign; the derivative of the numerator must be multiplied by the denominator first.
If the denominator is a single term (monomial), it is often easier to simplify the expression by dividing every term in the numerator by the denominator before differentiating with the power rule, rather than using the quotient rule.

The Chain Rule is used for differentiating composite functions (functions inside other functions), such as $y = f(g(x))$ .
The formula in Leibniz notation is: * $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$
Alternatively, in prime notation: * $\frac{d}{dx}[f(g(x))] = f'(g(x)) \times g'(x)$
The process involves taking the derivative of the "outer" function while keeping the "inner" function the same, then multiplying by the derivative of the "inner" function.
The General Power Rule is a specific case of the chain rule: * $\frac{d}{dx}([u(x)]^n) = n[u(x)]^{n-1} \times u'(x)$

Implicit differentiation is used when an equation is not solved explicitly for $y$ (e.g., $x^2 + y^2 = 25$ ) and it is difficult or impossible to isolate $y$ .
The procedure involves: 1. Differentiating both sides of the equation with respect to $x$ . 2. Treating $y$ as a function of $x$ . Whenever you differentiate a term containing $y$ , you must multiply by $\frac{dy}{dx}$ (due to the chain rule). 3. Collecting all terms containing $\frac{dy}{dx}$ on one side of the equation. 4. Factoring out $\frac{dy}{dx}$ . 5. Solving for $\frac{dy}{dx}$ .
Example: Differentiating $y^2 = x$ : * $\frac{d}{dx}(y^2) = \frac{d}{dx}(x)$ * $2y \times \frac{dy}{dx} = 1$ * $\frac{dy}{dx} = \frac{1}{2y}$

Related rates problems involve finding the rate at which one quantity changes by relating it to other quantities whose rates of change are known.
General steps for solving related rates problems: 1. Draw a diagram if applicable. 2. List the given information and the rate to be found (e.g., "Given $\frac{dr}{dt} = 2\,cm/s$ , find $\frac{dV}{dt}$ when $r = 5\,cm$ "). 3. Write an equation that relates the variables involved (e.g., volume of a sphere $V = \frac{4}{3}\pi r^3$ or the Pythagorean theorem $a^2 + b^2 = c^2$ ). 4. Differentiate both sides of the equation with respect to time ( $t$ ) using the chain rule/implicit differentiation. 5. Substitute the known values and solve for the unknown rate.
Important distinction: Never substitute the "snapshot" values (like "when $r = 5$ ") into the equation before differentiating; constants that do not change over time should be substituted before, but variables must remain variables until after differentiation.

Critical Points: A value $c$ in the domain of $f$ is a critical point if $f'(c) = 0$ or $f'(c)$ is undefined.
Extreme Value Theorem: If $f$ is continuous on a closed interval $[a, b]$ , then $f$ has both an absolute maximum and an absolute minimum on that interval.
Relative (Local) Extrema: * A relative maximum occurs if $f(x)$ changes from increasing (f'(x) > 0) to decreasing ( $f'(x) < 0$ ) at a critical point. * A relative minimum occurs if $f(x)$ changes from decreasing ( $f'(x) < 0$ ) to increasing ( $f'(x) > 0$ ) at a critical point.
The First Derivative Test: 1. Find all critical points. 2. Set up test intervals on a number line using the critical points. 3. Choose a test value in each interval and plug it into $f'(x)$ to determine the sign (positive for increasing, negative for decreasing). 4. Identify extrema based on sign changes.

The Mean Value Theorem states that for a function that meets specific criteria, there exists at least one point where the instantaneous rate of change equals the average rate of change over an interval.
Requirements for MVT: 1. The function $f(x)$ must be continuous on the closed interval $[a, b]$ . 2. The function $f(x)$ must be differentiable on the open interval $(a, b)$ .
The Theorem Statement: * There exists at least one number $c$ in $(a, b)$ such that: * $f'(c) = \frac{f(b) - f(a)}{b - a}$
Rolle’s Theorem is a special case of MVT where $f(a) = f(b)$ , implying that there is at least one point $c$ where the derivative is zero ( $f'(c) = 0$ ).