Derivative Rules and Techniques

Need to find derivatives quickly? These essential formulas will save you time on tests! The derivative of e^x is simply e^x itself—one of the coolest properties in calculus. When dealing with expressions like e^$x+4$, use the chain rule: e^$x+4$ = e^4·e^x, so its derivative is e^4·e^x.

For logarithms, remember that the derivative of ln x is 1/x. When you have more complex expressions like ln(x²), use the chain rule: d/dx[ln(x²)] = (2x)/(x²) = 2/x. Similarly, for logarithms with different bases, use the formula d/dx$log_a u$ = (1)/(u ln$a$)·du/dx.

Trigonometric functions follow their own patterns: the derivative of sin x is cos x, and the derivative of cos x is -sin x. For exponential functions with different bases, like 4^x, the derivative is 4^x·ln(4).

Pro Tip: When solving derivative problems, always identify which rule applies before starting. Most mistakes happen when students apply the wrong formula!

Basic logarithm properties are also important to remember:

• log(xy) = log x + log y
• log(x/y) = log x - log y
• log x^n = n log x

Finding Critical Points and Analyzing Function Behavior

Ever wonder how to find where a function peaks or dips? Critical points are where f'$x$ = 0 or f'$x$ is undefined—they're the key to finding important features on graphs! To determine where a function increases or decreases, analyze the sign of the first derivative.

When f'$x$ > 0, the function is increasing. When f'$x$ < 0, the function is decreasing. For example, to analyze y = -x⁴ - 4x³ + 20x² - 10, find f'$x$ = -4x³ - 12x² + 40x, then locate where this equals zero: x = -5, 0, and 2. Testing points in each interval reveals where the function increases and decreases.

Concavity tells us about the shape of the graph. When f''$x$ > 0, the function is concave up (smiles), and when f''$x$ < 0, it's concave down (frowns). A point of inflection occurs where concavity changes—this happens when f''$x$ = 0 and changes sign.

Remember: When analyzing functions, create a number line with your critical points and test the sign of f'$x$ in each interval—this visual approach helps avoid confusion!

For trigonometric functions like f$x$ = 12 + 4x - 3tan x, the same principles apply, but you'll need to work with the appropriate domain restrictions $like (0,π/2)$ and use trigonometric identities to solve.

Relative Extrema and Motion Applications

Finding where functions reach their highest or lowest points is crucial for many real-world problems! Relative extrema occur at critical points where the function changes direction. A relative maximum happens when f'$x$ changes from positive to negative, and a relative minimum occurs when f'$x$ changes from negative to positive.

You can confirm extrema using the second derivative test: if f'$x$ = 0 and f''$x$ < 0, you've found a relative maximum; if f''$x$ > 0, it's a relative minimum. For example, with y = x⁴ - 2x² + 8, find critical points at x = 0, ±1, then use f''$x$ to classify each point.

Motion applications connect derivatives to the physical world. Position (where you are) differentiates to velocity (how fast you're moving with direction), which differentiates to acceleration (how velocity changes). Remember:

• Positive velocity means moving forward/up/right
• Negative velocity means moving backward/down/left
• Positive acceleration means velocity is increasing
• Negative acceleration means velocity is decreasing

Physics Connection: When velocity and acceleration have the same sign, speed is increasing. When they have opposite signs, speed is decreasing. This helps analyze real-world motion problems!

Implicit differentiation helps when you can't easily solve for y. Differentiate both sides with respect to x, remembering to use the chain rule for y-terms (multiply by dy/dx). This technique helps find equations of tangent lines and identify horizontal and vertical tangents.