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 for differentiating 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 a function is nested inside another function.

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

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

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

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

Differentiation of Exponential and Logarithmic Functions

Exponential and logarithmic functions are differentiated using their unique properties and, often, the Chain Rule.

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

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

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

Implicit Differentiation and Tangent Lines

Implicit differentiation is used when a function is not given explicitly in terms of one variable. It is also useful for finding the equation of the tangent line to a curve at a given point.

• Implicit Differentiation: Differentiate both sides of the equation with respect to x, treating y as a function of x.

• Example: For $x^2 + y^2 = 25$, $2x + 2y \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}$

• Tangent Line Equation: At point $(3, 4)$, slope $m = -\frac{3}{4}$, so the tangent line is $y - 4 = -\frac{3}{4}(x - 3)$

Finding Tangent Lines to Parametric Curves

To find the equation of the tangent line to a curve defined by a function at a given point, differentiate the function and evaluate the derivative at the specified point.

• Example: For $y = \sin x \cos x$, $y' = \cos x \cos x + (-\sin x) \sin x = \cos^2 x - \sin^2 x$

• At $x = \pi$: $y' = 1$

• Tangent Line: Use the point-slope form with the calculated slope.