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

When differentiating functions of the form $y = [U(x)]^n$, where $U(x)$ is a differentiable function and $n$ is a real number, the power rule can be combined with the chain rule:

• 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 Functions

Exponential functions can be differentiated using the chain rule and the properties of exponents:

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

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

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

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

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 $y'(C) = 1$

• Application: Used to find tangent lines to curves defined by explicit or implicit equations.

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 $\frac{dy}{dx}$, differentiate both sides with respect to $x$ and solve for $\frac{dy}{dx}$:

• Example: If $x^2 + y^2 = 25$, then $2x + 2y \frac{dy}{dx} = 0$ leads to $\frac{dy}{dx} = -\frac{x}{y}$

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