Techniques of Differentiation: The Chain Rule and Applications

Techniques of Differentiation

The Chain Rule

The Chain Rule is a fundamental technique in calculus for finding the derivative of composite functions. If a function is composed of two differentiable functions, the derivative of the composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.

• Definition: If g is differentiable at x and f is differentiable at g(x), then the composite function F(x) = f(g(x)) is differentiable at x and its derivative is given by:

$F'(x) = f'(g(x)) \cdot g'(x)$

• Example: For $F(x) = (2x^2 + 1)^{100}$, the derivative is:

$F'(x) = 100 \left(2x^2 + 1\right)^{99} \cdot 4x = 400x \left(2x^2 + 1\right)^{99}$

The Power Rule Combined with the Chain Rule

When differentiating a power of a function, the Power Rule can be combined with the Chain Rule. If $U = g(x)$ is differentiable and $n$ is any real number, then:

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

• Example: Differentiate $y = (x^2 - 1)^{100}$:

$y' = 100(x^2 - 1)^{99} \cdot 2x = 200x(x^2 - 1)^{99}$

• Exponential Functions: For $f(x) = e^{u}$, $f'(x) = e^{u} \cdot u'$

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

Applications: Tangent Lines and Implicit Differentiation

Finding the equation of the tangent line to a curve at a given point is a common application of differentiation. This often involves evaluating the derivative at a specific point and using the point-slope form of a line.

• Example: Find the equation of the tangent line to $y = \sin x \cos x$ at $x = \pi$.

• First, compute the derivative using the product rule:

$y' = \cos x \cos x + \sin x (-\sin x) = \cos^2 x - \sin^2 x$

• Evaluate at $x = \pi$:

$y'(\pi) = \cos^2 \\pi - \sin^2 \\pi = 1 - 0 = 1$

• Use the point $(\pi, 0)$ and the slope $1$ to write the tangent line:

$y - 0 = 1(x - \pi) \implies y = x - \pi$

Implicit Differentiation and Tangent Lines to Circles

Implicit differentiation is used when a function is not given explicitly as $y = f(x)$, but rather as a relation involving both $x$ and $y$. To find $\frac{dy}{dx}$, differentiate both sides with respect to $x$, treating $y$ as a function of $x$.

• Example: For the circle $x^2 + y^2 = 25$, differentiate both sides:

$2x + 2y \frac{dy}{dx} = 0$

$\frac{dy}{dx} = -\frac{x}{y}$

• At the point $(3, 4)$, the slope is $-\frac{3}{4}$.

• The equation of the tangent line at $(3, 4)$ is:

$y - 4 = -\frac{3}{4}(x - 3)$

Further Examples: Exponential and Logarithmic Differentiation

Exponential and logarithmic functions often require the use of the Chain Rule and properties of logarithms for differentiation.

• Example: $f(x) = e^{-2x + 5x}$

$f'(x) = e^{-2x + 5x} \cdot (-2 + 5) = 3e^{-2x + 5x}$

• General Rule: For $y = b^x$, $\frac{d}{dx} b^x = b^x \ln b$

• Example: If $g(x) = 2^x$, then $g'(x) = 2^x \ln 2$