Techniques of Differentiation: 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 = f \circ g is differentiable, and its derivative is given by:

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

• Application: Used when differentiating functions nested within other functions.

• Example: If $F(x) = (2x^2 + 1)^{100}$, then $F'(x) = 100 \left(2x^2 + 1\right)^{99} \cdot 4x$

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

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

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

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

Differentiation of Exponential and Logarithmic Functions

Exponential and logarithmic functions have specific differentiation rules, often used in combination with the Chain Rule.

• Exponential Rule: $\frac{d}{dx} \left( e^{u(x)} \right) = e^{u(x)} \cdot u'(x)$

• Logarithmic Rule: $\frac{d}{dx} \left( b^{x} \right) = b^{x} \ln b$

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

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

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.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $m$ is the slope at $(x_1, y_1)$

• Example: For $y = \sin x \cos x$ at $x = \pi$, $y' = \cos x \cos x + (-\sin x)(\sin x)$, and $y'( ext{at } x = \pi) = 1$

• Application: Used to find tangent lines to curves at specific points.

Implicit Differentiation and Tangent Lines to Implicit Curves

Implicit differentiation is used when a function is not given explicitly in terms of $y = f(x)$, but rather as an equation involving both $x$ and $y$. The derivative $\frac{dy}{dx}$ is found by differentiating both sides with respect to $x$ and solving for $\frac{dy}{dx}$.

• Implicit Differentiation: For $x^2 + y^2 = 25$, differentiate both sides to get $2x + 2y \frac{dy}{dx} = 0$

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

• Tangent Line at (3,4): Slope $m = -\frac{3}{4}$, equation $y - 4 = -\frac{3}{4}(x - 3)$

Summary Table: Differentiation Techniques

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