The derivative gives the instantaneous rate of change of a function at a specific point, which is interpreted as the slope of the tangent line. This concept is fundamental in understanding rates and slopes in calculus.

The standard definition of the derivative uses the limit as h approaches 0 of the difference quotient [f(a+h) - f(a)] / h. This definition captures the instantaneous rate of change of the function at x = a.

Differentiability at a point implies continuity at that point, though the converse is not necessarily true. This is a key concept in understanding the behavior of functions in calculus.

The Power Rule states that the derivative of x❿ is n⋅x❿❻¹, which is the fundamental rule used for differentiating power functions. This rule is one of the first differentiation techniques learned in calculus.

The derivative of a function at a specific point provides the slope of the tangent line, which is then used with the point-slope formula to determine the equation of the tangent line. This approach is central to differential calculus.

Using the power rule, the derivative of 3x❴ is computed as 3*4*x³, which simplifies to 12x³. This is a straightforward application of the rule.

Differentiating term-by-term, the derivative of 5x³ is 15x², the derivative of -2x is -2, and the constant 7 differentiates to 0. Thus, f'(x) = 15x² - 2.

By applying the quotient rule, the derivative is calculated as [x*(4x) - (2x² + 3)*(1)] / x², which simplifies to (2x² - 3)/x². This method is essential for derivatives of rational functions.

Composite functions require the Chain Rule because the function sin(x²) is composed of the outer function sin(u) and the inner function u = x². The Chain Rule systematically differentiates such compositions.

The exponential function eˣ has the unique property that its derivative is itself, making the differentiation process straightforward. This characteristic is a hallmark of the exponential function.

For x > 0, the natural logarithm function ln(x) differentiates to 1/x according to standard logarithmic differentiation rules. This derivative is fundamental in calculus.

Using the product rule, the derivative of f(x)·g(x) is f'(x)·g(x) + f(x)·g'(x), which computes to 2x·3x + x²·3 = 9x². This rule is central when dealing with products of functions.

The first derivative of x³ is 3x²; differentiating this once more yields the second derivative, 6x. This process shows how successive differentiation works.

Expressing √x as x^(1/2) allows the use of the power rule. Differentiation then gives (1/2)x^(1/2-1), which simplifies to (1/2)x^(-1/2).

By applying the chain rule, the derivative of cos(3x) is -sin(3x) multiplied by the derivative of 3x (which is 3), yielding -3sin(3x). This illustrates the use of the chain rule in trigonometric functions.

A local maximum at x = c is typically confirmed by first finding f'(c) = 0 and then using the second derivative test; if f''(c) < 0, the function is concave down and has a local maximum. This condition is a standard criterion in calculus.

Despite the oscillatory behavior of sin(1/x), the factor x² dampens the oscillations as x approaches 0, allowing the derivative at 0 to exist and equal 0 by the Squeeze Theorem. This is a classic example demonstrating differentiability at a point where the function oscillates.

Using the product rule, differentiate x and e^(x²) separately. The derivative of e^(x²) requires the chain rule, yielding 2x·e^(x²). Combining these results produces e^(x²)(1 + 2x²).

The derivative of the inverse function is given by (f❻¹)'(y) = 1 / f'(x) where x = f❻¹(2). Solving x³ + 1 = 2 gives x = 1, and f'(1) = 3, so the derivative is 1/3.

When f''(a) equals zero, the second derivative test does not provide enough information to conclude whether f has a local maximum or minimum at x = a. Additional methods, such as the first derivative test or higher-order derivative tests, are required.