Differentiation Techniques and Applications: Chain Rule, Power Rule, and Tangent Lines

Techniques of Differentiation

The Chain Rule

The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. If g is differentiable and f is differentiable at g(x), then the composite function F(x) = f(g(x)) is differentiable, and its derivative is given by:

• Formula: $F'(x) = f'(g(x)) \cdot g'(x)$

• Application: Used when functions are nested, such as $f(x) = (2x^2 + 1)^{100}$.

• Example: For $F(x) = (2x^2 + 1)^{100}$, $F'(x) = 100 \cdot (2x^2 + 1)^{99} \cdot 4x$.

The Power Rule Combined with the Chain Rule

The Power Rule can be extended using the Chain Rule for functions of the form $y = [U(x)]^n$, where $U(x)$ is a differentiable function and $n$ is a real number.

• Formula: $\frac{d}{dx} [U^n] = n U^{n-1} \frac{dU}{dx}$

• Example: For $y = (x^2 - 100)^{99}$, $y' = 99(x^2 - 100)^{98} \cdot 2x$.

• Exponential Functions: If $f(x) = e^{U}$, then $f'(x) = e^{U} \frac{dU}{dx}$.

• Example: $y = e^{\sin x}$, $y' = e^{\sin x} \cos x$.

Applications of Derivatives

Finding Tangent Lines to Curves

To find the equation of the tangent line to a curve at a given point, first compute the derivative to find the slope at that point, then use the point-slope form of a line.

• Example: For $y = \sin x \cos x$ at $x = \pi$, the derivative is $y' = \cos x \cos x + (-\sin x) \sin x = \cos^2 x - \sin^2 x$.

• At $x = \pi$, $y' = 1$ and the tangent line equation is $y - (-1) = 1(x - \pi)$.

Implicit Differentiation and Tangent Lines

Implicit differentiation is used when a function is not given explicitly in terms of one variable. To find the tangent line to a curve defined implicitly, differentiate both sides with respect to $x$, solve for $\frac{dy}{dx}$, and use the point-slope form.

• Example: For $x^2 + y^2 = 25$, differentiate to get $2x + 2y \frac{dy}{dx} = 0$.

• Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{x}{y}$.

• At the point $(3,4)$, the slope is $-\frac{3}{4}$, and the tangent line is $y - 4 = -\frac{3}{4}(x - 3)$.

Derivatives of Exponential Functions

Differentiating Exponential Functions

Exponential functions of the form $f(x) = e^{u(x)}$ are differentiated using the Chain Rule. For general bases, the derivative of $b^x$ is $b^x \ln b$.

• Formula: $\frac{d}{dx} (b^x) = b^x \ln b$

• Example: $f(x) = e^{-2x + 5x}$, $f'(x) = (-2x + 5) e^{-2x + 5x}$

• Example: $g(x) = 4^x$, $g'(x) = 4^x \ln 4$

Summary Table: Key Differentiation Rules

Rule	Formula	Example
Chain Rule	$F'(x) = f'(g(x)) \cdot g'(x)$	$F(x) = (2x^2 + 1)^{100}$
Power Rule (with Chain Rule)	$\frac{d}{dx} [U^n] = n U^{n-1} \frac{dU}{dx}$	$y = (x^2 - 100)^{99}$
Exponential Rule	$\frac{d}{dx} (e^{u(x)}) = e^{u(x)} \frac{du}{dx}$	$y = e^{\sin x}$
General Exponential	$\frac{d}{dx} (b^x) = b^x \ln b$	$y = 4^x$
Implicit Differentiation	$\frac{dy}{dx} = -\frac{x}{y}$	$x^2 + y^2 = 25$