91Ó°ÊÓ

Trigonometric functions play a crucial role in calculus, and understanding their derivatives is key. Familiar trigonometric functions include sine, cosine, tangent, cotangent, secant, and cosecant. Each has its own unique derivative:

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).
• The derivative of \(\tan x\) is \(\sec^2 x\).
• The derivative of \(\cot x\) is \(-\csc^2 x\).
• The derivative of \(\sec x\) is \(\sec x \tan x\).
• The derivative of \(\csc x\) is \(-\csc x \cot x\).

Knowing these derivatives allows you to tackle complex calculus problems involving these functions. Recognizing how they change with respect to \(x\) is essential, especially when integrated into larger expressions or equations.

The chain rule is a fundamental technique for finding the derivative of composite functions. It is used when a function is applied inside another function, commonly in the form \(f(g(x))\). The chain rule states that:\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\]This means you first differentiate the outer function \(f\) with respect to \(g(x)\), then multiply by the derivative of the inner function \(g(x)\).

Example of Chain Rule

Suppose you have \(f(x) = \sec(x^2)\). Here, the outer function is \(\sec(u)\) where \(u = x^2\). Using the chain rule:

• Find \(\frac{d}{du}[\sec u] = \sec u \tan u\).
• Find \(\frac{d}{dx}[x^2] = 2x\).
• Multiply: \(\frac{d}{dx}[\sec(x^2)] = \sec(x^2) \tan(x^2) \cdot 2x\).

The chain rule is vital in calculus, especially for functions where operations are nested within each other.

The quotient rule is essential when differentiating functions that are ratios of two differentiable functions, in the form \(\frac{u(x)}{v(x)}\). The quotient rule formula is:\[\frac{d}{dx} \left[\frac{u(x)}{v(x)}\right] = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}\]This rule effectively gives you the derivative of a fraction, ensuring you're considering the change in both the numerator and the denominator.

Example of Quotient Rule

Suppose you need to find the derivative of \( \frac{\cos x}{\sin x} \). Using the quotient rule:

• Let \(u(x) = \cos x\) and \(v(x) = \sin x\).
• Derivative \(u'(x) = -\sin x\) and \(v'(x) = \cos x\).
• Apply the formula: \(\frac{d}{dx}\left[\frac{\cos x}{\sin x}\right] = \frac{\sin x \cdot (-\sin x) - \cos x \cdot \cos x}{\sin^2 x}\).
• Simplify the result to obtain \(-\csc^2 x\).

Mastering the quotient rule aids significantly in handling divisions within calculus problems.