91Ó°ÊÓ

The chain rule is an essential differentiation technique when dealing with composite functions. It allows us to differentiate the outer function first and then multiply it by the derivative of the inner function. This rule is crucial when we have functions nested within each other, like in our example where we have a cosine function containing a logarithmic function.
The chain rule formula is usually articulated as:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]
By applying this rule, you can tackle complex differentiation problems with ease. More precisely, it enables step-by-step processing of each function layer. Make sure to correctly identify the outer and inner functions.
In our problem, we differentiate \[ g(t) = \cos(\ln(t)) \] by breaking it down into its components.

Differentiating complex functions often requires a good grasp of several techniques. Here, we focus on the necessity to handle both elementary and composite functions. Common techniques include:

• Basic rule for polynomials: Power rule
• Product rule for products of functions: Chain Rule
• Quotient rule for ratios of functions: Quotient Rule
• Trigonometric functions: Sine and Cosine rules
• Logarithmic functions: Logarithm rule

In our example, we combined trigonometric differentiation with natural logarithms. Differentiating the outer cosine function results in:
\[ \frac{d}{du}[\cos(u)] = -\sin(u) \]
Then, differentiate the inner logarithmic function:
\[ \frac{d}{dt}[\ln(t)] = \frac{1}{t} \]
Finally, apply the chain rule to combine these derivatives.

Composite functions are formed when one function is nested inside another. Understanding them is vital for effectively applying the chain rule.
In the problem given, we identify the composite function: \[ g(t) = \cos(\ln(t)) \]
Here, \[ \cos(u) \] is the outer function and \[ \ln(t) \] is the inner function. To differentiate composite functions, we need to:

• Identify the outer and inner functions accurately.
• Differentiate the outer function with respect to the inner variable.
• Differentiate the inner function with respect to the main variable.
• Multiply both derivatives according to the chain rule.

So, integrating these steps for \[ g(t) = \cos(\ln(t)) \], we get:
\[ g'(t) = \frac{d}{du}[\cos(u)] \cdot \frac{d}{dt}[\ln(t)] = -\sin(\ln(t)) \cdot \frac{1}{t} = -\frac{\sin(\ln(t))}{t} \]