Solve the given problems. Find the slope of a line tangent...

Solve the given problems.
Find the slope of a line tangent to the curve of $y=\tan ^{-1} 2 x+\ln \left(4 x^{2}+1\right),$ where $x=0.625 .$ Verify the result
by using the numerical derivative feature of a calculator.

Key Concepts

Numerical Derivative

The numerical derivative approximates the derivative of a function by using finite differences. It is a practical method employed to verify analytical results, especially when the function is complex or when using computational tools like calculators that feature numerical derivative capabilities.

Differentiation of Inverse Trigonometric Functions

The differentiation of inverse trigonometric functions involves specific derivative formulas. For instance, the derivative of the inverse tangent function is used to handle parts of functions that involve inverse trigonometric expressions, which is essential when these components are part of a larger composite function.

Differentiation of Logarithmic Functions

The derivative of logarithmic functions is based on the property that the derivative of ln(u) is u'/u, where u is a function of the variable. This rule is especially useful when the logarithmic function's argument is more complex than a simple variable, requiring the application of the chain rule.

Tangent Line

A tangent line to a curve at a specific point is a straight line that touches the curve only at that point, having the same slope as the curve does at that point. This slope is found by evaluating the derivative of the function at the given location.

Derivative

The derivative of a function measures the instantaneous rate of change of the function with respect to a variable. It is central to understanding how a slight change in the input variable influences the output, which is mathematically represented by the slope of the tangent line to the curve at a given point.

Chain Rule

The chain rule is a fundamental technique in differentiation used for computing the derivative of a composite function. When a function is composed of two or more functions, the chain rule allows for the derivative of the outer function to be multiplied by the derivative of the inner function, simplifying complex differentiation problems.

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