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: Derivative of $\frac{\cos x}{x^2}$

• Given: $f(x) = \frac{\cos x}{x^2}$

• Apply the quotient rule:

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

• Simplified:

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

• Further Example: $f(x) = \frac{2x \cos^2(x)}{(x^2+1) \cos(x)}$

• Derivative:

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

• This uses both the product and quotient rules, as well as trigonometric identities.

Derivatives of Logarithmic and Power Functions

Power Rule and Logarithmic Differentiation

The power rule states that for $y = x^n$, the derivative is:

$\frac{d}{dx} x^n = n x^{n-1}$

For logarithmic functions, the derivative of $y = \ln(x)$ is:

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

• Logarithmic differentiation is useful for functions involving products, quotients, or powers of variable expressions.

Example: Derivative of $y = 3/5 x^{5/2} + 3/5 x^{1/2} + \ln(x^5) + 3 x^4 + x^n$

• Apply the power rule and logarithmic differentiation:

$y' = 3/5 \cdot \frac{5}{2} x^{3/2} + 3/5 \cdot \frac{1}{2} x^{-1/2} + \frac{1}{x^5} \cdot 5x^4 + 12x^3 + n x^{n-1}$

• Simplified:

$y' = \frac{3}{2} x^{3/2} + \frac{3}{10} x^{-1/2} + 5 \ln(x) + 12x^3 + n x^{n-1}$

• For $\ln(x^5)$, use the property $\ln(x^5) = 5 \ln(x)$.

Derivatives Involving Natural Logarithms

Derivative of $y = \ln(x^2 - x + 1)$

To differentiate $y = \ln(x^2 - x + 1)$, use the chain rule:

• Chain Rule: $\frac{d}{dx} \ln(u(x)) = \frac{1}{u(x)} \cdot u'(x)$

$y' = \frac{1}{x^2 - x + 1} \cdot (2x - 1)$

• Example: For $y = \ln(x^2 - x + 1)$, $u(x) = x^2 - x + 1$, $u'(x) = 2x - 1$.

Summary Table: Common Derivative Rules

Function	Derivative	Notes
$x^n$	$n x^{n-1}$	Power rule
$\ln(x)$	$\frac{1}{x}$	Logarithmic rule
$\sin(x)$	$\cos(x)$	Trigonometric rule
$\cos(x)$	$-\sin(x)$	Trigonometric rule
$\frac{g(x)}{h(x)}$	$\frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$	Quotient rule

Additional info:

• Some expressions in the original notes were incomplete or unclear; standard calculus rules and logical completion were used to fill gaps.

• Highlighted lines indicate important results or final simplified derivatives.