Calculus Study Notes: Derivatives of Quotients and Logarithmic Functions

Derivatives of Quotient Functions

Quotient Rule for Derivatives

The quotient rule is used to find the derivative of a function that is the ratio of two differentiable functions. If $f(x) = \frac{g(x)}{h(x)}$, then the derivative is given by:

• Formula:

$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$

• Key Points:

• Differentiate the numerator and denominator separately.

• Subtract the product of the numerator and the derivative of the denominator from the product of the derivative of the numerator and the denominator.

• Divide the result by the square of the denominator.

Example: Find the derivative of $f(x) = \frac{\cos x}{x^2}$.

$f'(x) = \frac{-\sin x \cdot x^2 - \cos x \cdot 2x}{(x^2)^2}$

Simplified:

$f'(x) = \frac{-x^2 \sin x - 2x \cos x}{x^4}$

Quotient Rule with Trigonometric Functions

When applying the quotient rule to trigonometric functions, use the derivatives of sine and cosine:

• $\frac{d}{dx}(\sin x) = \cos x$

• $\frac{d}{dx}(\cos x) = -\sin x$

Example: Find the derivative of $f(x) = \frac{2x \cos^2(x)}{(x^2+1) \cos(x)}$.

$f'(x) = \frac{2x \cos^2(x) \cdot \frac{d}{dx}[(x^2+1) \cos(x)] - (x^2+1) \cos(x) \cdot \frac{d}{dx}[2x \cos^2(x)]}{[(x^2+1) \cos(x)]^2}$

Simplified (as highlighted in the notes):

$f'(x) = \frac{2x \cos^2(x) + (x^2+1) \sin(x) \cos(x)}{(x^2+1) \cos(x)}$

Derivatives of Logarithmic and Power Functions

Derivative of Logarithmic Functions

The derivative of the natural logarithm function $\ln(x)$ is:

• $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$

For more complex logarithmic expressions, use the chain rule and properties of logarithms.

Derivative of Power Functions

The derivative of a power function $x^n$ is:

• $\frac{d}{dx}[x^n] = n x^{n-1}$

Worked Example: Derivative of a Sum Involving Powers and Logarithms

Given $y = \frac{3}{5}x^{5/2} + 3x^{1/5} + \ln(x^5) + 3x^2 + \ln(5)$, find $\frac{dy}{dx}$.

• Differentiate each term separately:

• $\frac{d}{dx}[\frac{3}{5}x^{5/2}] = \frac{3}{5} \cdot \frac{5}{2} x^{3/2} = \frac{3}{2} x^{3/2}$

• $\frac{d}{dx}[3x^{1/5}] = 3 \cdot \frac{1}{5} x^{-4/5} = \frac{3}{5} x^{-4/5}$

• $\frac{d}{dx}[\ln(x^5)] = \frac{d}{dx}[5 \ln(x)] = \frac{5}{x}$

• $\frac{d}{dx}[3x^2] = 6x$

• $\frac{d}{dx}[\ln(5)] = 0$ (since $\ln(5)$ is a constant)

Final Answer:

$\frac{dy}{dx} = 3x^{3/2} + \frac{3}{5}x^{-4/5} + \frac{5}{x} + 6x$

Logarithmic Differentiation

Using Logarithmic Properties for Differentiation

Logarithmic differentiation is useful for functions involving products, quotients, or powers. The properties of logarithms help simplify differentiation:

• $\ln(ab) = \ln(a) + \ln(b)$

• $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$

• $\ln(a^n) = n \ln(a)$

Example: For $y = \ln(x^2 - x + 1)$, the derivative is:

$\frac{d}{dx}[\ln(x^2 - x + 1)] = \frac{2x - 1}{x^2 - x + 1}$

Summary Table: Common Derivative Rules

Function	Derivative
$x^n$	$n x^{n-1}$
$\ln(x)$	$\frac{1}{x}$
$\sin(x)$	$\cos(x)$
$\cos(x)$	$-\sin(x)$
$\frac{g(x)}{h(x)}$	$\frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$

Additional info: Some steps and simplifications were inferred from context and standard calculus rules, as the original notes contained shorthand and partial calculations.