Differentiation Cheat Sheet

Cheatsheet Content

Definition of Derivative Limit Definition: The derivative of a function $f(x)$ with respect to $x$ is: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ Alternative Form: $$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$ Notation: $f'(x)$, $\frac{dy}{dx}$, $\frac{d}{dx}[f(x)]$, $y'$ Basic Differentiation Rules Constant Rule: $\frac{d}{dx}[c] = 0$ Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$ Constant Multiple Rule: $\frac{d}{dx}[cf(x)] = c f'(x)$ Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$ Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$ Derivatives of Trigonometric Functions $\frac{d}{dx}[\sin x] = \cos x$ $\frac{d}{dx}[\cos x] = -\sin x$ $\frac{d}{dx}[\tan x] = \sec^2 x$ $\frac{d}{dx}[\cot x] = -\csc^2 x$ $\frac{d}{dx}[\sec x] = \sec x \tan x$ $\frac{d}{dx}[\csc x] = -\csc x \cot x$ Derivatives of Inverse Trigonometric Functions $\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}[\arccos x] = -\frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}$ $\frac{d}{dx}[\text{arccot } x] = -\frac{1}{1+x^2}$ $\frac{d}{dx}[\text{arcsec } x] = \frac{1}{|x|\sqrt{x^2-1}}$ $\frac{d}{dx}[\text{arccsc } x] = -\frac{1}{|x|\sqrt{x^2-1}}$ Derivatives of Exponential and Logarithmic Functions $\frac{d}{dx}[e^x] = e^x$ $\frac{d}{dx}[a^x] = a^x \ln a$ $\frac{d}{dx}[\ln x] = \frac{1}{x}$ $\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$ Higher-Order Derivatives Second Derivative: $\frac{d^2y}{dx^2} = f''(x)$ Third Derivative: $\frac{d^3y}{dx^3} = f'''(x)$ N-th Derivative: $\frac{d^ny}{dx^n} = f^{(n)}(x)$ Implicit Differentiation Used when $y$ is not explicitly defined as a function of $x$. Differentiate both sides of the equation with respect to $x$. Remember to apply the Chain Rule to terms involving $y$, e.g., $\frac{d}{dx}[y^n] = ny^{n-1}\frac{dy}{dx}$. Solve for $\frac{dy}{dx}$. Example: Given $x^2 + y^2 = 25$ $2x + 2y\frac{dy}{dx} = 0$ $2y\frac{dy}{dx} = -2x$ $\frac{dy}{dx} = -\frac{x}{y}$ Related Rates Identify all given rates and the rate to be found. Write an equation relating the quantities involved. Differentiate the equation with respect to time ($t$) using the Chain Rule. Substitute known values and solve for the unknown rate. Applications of Derivatives Slope of a Tangent Line: $f'(a)$ is the slope of the tangent line to $f(x)$ at $x=a$. Equation of Tangent Line: $y - f(a) = f'(a)(x - a)$ Critical Points: Points where $f'(x) = 0$ or $f'(x)$ is undefined. Increasing/Decreasing Functions: $f'(x) > 0 \implies f(x)$ is increasing. $f'(x) Local Extrema (First Derivative Test): If $f'(x)$ changes from $+$ to $-$ at $c$, $f(c)$ is a local maximum. If $f'(x)$ changes from $-$ to $+$ at $c$, $f(c)$ is a local minimum. Concavity (Second Derivative): $f''(x) > 0 \implies f(x)$ is concave up. $f''(x) Inflection Points: Points where concavity changes ($f''(x) = 0$ or undefined and $f''(x)$ changes sign). Local Extrema (Second Derivative Test): If $f'(c) = 0$ and $f''(c) > 0$, $f(c)$ is a local minimum. If $f'(c) = 0$ and $f''(c) If $f'(c) = 0$ and $f''(c) = 0$, test is inconclusive (use First Derivative Test). Optimization: Find absolute maximum/minimum values on an interval using critical points and endpoints. L'Hôpital's Rule: If $\lim_{x \to c} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$ Linear Approximation: $L(x) = f(a) + f'(a)(x-a)$ for $x$ near $a$. Differentials: $dy = f'(x)dx$ (change in $y$ along the tangent line) Common Derivatives Summary Table Function $f(x)$ Derivative $f'(x)$ $c$ (constant) $0$ $x^n$ $nx^{n-1}$ $e^x$ $e^x$ $a^x$ $a^x \ln a$ $\ln x$ $1/x$ $\log_a x$ $1/(x \ln a)$ $\sin x$ $\cos x$ $\cos x$ $-\sin x$ $\tan x$ $\sec^2 x$ $\sec x$ $\sec x \tan x$ $\arcsin x$ $1/\sqrt{1-x^2}$ $\arctan x$ $1/(1+x^2)$