Calculus Study Notes: Derivatives, Tangents, and Applications

Derivatives and Their Applications

Finding Derivatives

Derivatives measure the rate at which a function changes with respect to its variable. They are fundamental in calculus for analyzing functions, motion, and change.

• Definition: The derivative of a function y = f(x) with respect to x is denoted as dy/dx or f'(x).

• Higher-Order Derivatives: The second derivative is d2y/dx2, and the third derivative is d3y/dx3.

• Example: For y = x^n, the first derivative is $\frac{dy}{dx} = nx^{n-1}$.

Sample Problems

• Find dy/dx for: y = 2x / x2 and y = x e^{\sqrt{x^2 + 1}}

• Find d3x/dt3 for: y = \log_2 x

• Find dy/dx for: y = x^4 and y = x^3 + y^3 = 4x^2y

Implicit Differentiation

When a function is defined implicitly (not solved for y in terms of x), use implicit differentiation to find derivatives.

• Example: For x^3 + y^3 = 4x^2y, differentiate both sides with respect to x, treating y as a function of x.

Motion Along a Line: Position, Velocity, and Acceleration

In kinematics, the position, velocity, and acceleration of a body moving along a straight line are related through derivatives.

• Position: s(t) gives the location at time t.

• Velocity: $v(t) = \frac{ds}{dt}$

• Acceleration: $a(t) = \frac{d^2s}{dt^2}$

• Application: To find when a body changes direction, set v(t) = 0 and solve for t.

Simplifying Expressions Involving Derivatives

Some problems require simplifying expressions after differentiation, often involving trigonometric, exponential, or logarithmic functions.

• Example: Simplify $y = \sin^{-1}(e^{-x}) + \sec^{-1}(e^x)$ for x ≥ 0.

Equations of Tangent Lines

The tangent line to a curve at a point gives the best linear approximation to the curve at that point.

• Equation of Tangent Line: $y - y_0 = f'(x_0)(x - x_0)$, where (x_0, y_0) is the point of tangency.

• Example: Find the tangent to y = x^4 + \ln x at x = e.

• Example: Find the tangent to y = \sin(\arccos 2x) at x = 1/4.

Definition of the Derivative (Limit of Difference Quotient)

The derivative at a point can be defined as the limit of the difference quotient:

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

• Application: Use this definition to find the derivative of f(x) = 1 + x.

Inverse Functions and Their Derivatives

If a function f has an inverse, the derivative of the inverse can be found using the formula:

• $\left( f^{-1} \right)'(y) = \frac{1}{f'(x)}$ where y = f(x)

• Application: If the graph of f passes through (1, 3) and has a slope of 1/2 there, the slope of the tangent to the inverse at x = 3 is 2.

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to another, often using the chain rule.

• Example: The volume of an expanding sphere is increasing at a rate of 16π ft3/h when r = 1 ft. Find the rate at which the radius is increasing.

• Formula for the volume of a sphere: $V = \frac{4}{3}\pi r^3$

• Differentiate both sides with respect to time: $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$