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 a function is nested inside another function.

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

Power Rule Combined with the Chain Rule

The Power Rule can be combined with the Chain Rule to differentiate 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: $y = (2x^2 + 1)^{100}$, $y' = 100 (2x^2 + 1)^{99} \cdot 4x$

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

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

Differentiation of Exponential and Logarithmic Functions

Exponential and logarithmic functions have specific differentiation rules. For $f(x) = e^{u(x)}$, the derivative is $f'(x) = e^{u(x)} \cdot u'(x)$. For $y = b^x$, the derivative is $y' = b^x \ln b$.

• Formula for Exponential: $\frac{d}{dx} [e^{u(x)}] = e^{u(x)} \cdot u'(x)$

• Formula for General Exponential: $\frac{d}{dx} [b^x] = b^x \ln b$

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

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

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.

• Step 1: Find $y'$ (the derivative) to get the slope.

• Step 2: Evaluate $y'$ at the given $x$ value.

• Step 3: Use the point-slope form: $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is the point.

• Example: For $y = \sin x \cos x$ at $x = \pi$, $y' = \cos x \cos x + (-\sin x)(\sin x)$, evaluated at $x = \pi$ gives $y' = 1$.

Implicit Differentiation and Tangent Lines

Implicit differentiation is used when a function is not given explicitly as $y = f(x)$, but rather as an equation involving both $x$ and $y$. To find the tangent line, differentiate both sides with respect to $x$, solve for $\frac{dy}{dx}$, and use the point-slope form.

• Step 1: Differentiate both sides of the equation with respect to $x$.

• Step 2: Solve for $\frac{dy}{dx}$.

• Step 3: Substitute the given point to find the slope.

• Step 4: Use the point-slope form to write the tangent line equation.

• Example: For $x^2 + y^2 = 25$ at $(3, 4)$, $\frac{dy}{dx} = -\frac{x}{y}$, so at $(3, 4)$, $\frac{dy}{dx} = -\frac{3}{4}$.

Summary Table: Differentiation Rules

Rule	Formula	Example
Chain Rule	$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$	$\frac{d}{dx} (2x^2 + 1)^{100}$
Power Rule	$\frac{d}{dx} x^n = n x^{n-1}$	$\frac{d}{dx} x^5 = 5x^4$
Exponential Rule	$\frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x)$	$\frac{d}{dx} e^{\sin x} = e^{\sin x} \cos x$
General Exponential	$\frac{d}{dx} b^x = b^x \ln b$	$\frac{d}{dx} 4^x = 4^x \ln 4$
Implicit Differentiation	$\frac{dy}{dx}$ from $F(x, y) = 0$	$\frac{dy}{dx} = -\frac{x}{y}$ for $x^2 + y^2 = 25$