Derivatives of Exponential Functions & Logarithmic Differentiation Calculus lnx, e^2x, x^x, x^sinx

Derivative of a^u (Constant Base, Variable Exponent)00:09:55

When dealing with a constant 'a' raised to a variable 'u', the derivative is a^u * u' * ln a. This is demonstrated with examples like 2^x, 4^(x^2), and 7^(4x - x^2).

Derivative of Natural Logarithm (ln u)00:13:02

The derivative of ln u is u' divided by u. This is applied to ln x, ln (x^2), ln (2x), ln (x+1), ln (x^2+1), ln (sin x), and ln (cos x).

Derivative of General Logarithm (log_a u)00:18:09

The derivative of log_a u is u' divided by (u * ln a). This formula is used for examples like log_3 (2x), log_5 (x^2), and log_7 (1-x).

Product Rule with Logarithms00:21:25

The video revisits the product rule (f'g + fg') to solve derivatives like x ln x, x^2 ln x, and x e^x, demonstrating how to combine different derivative rules.

Derivative of e^u00:00:23

The derivative of e raised to the power of u (where u is a function of x) is e^u multiplied by the derivative of u (u'). This is because the natural logarithm of e (ln e) is 1, simplifying the general formula for a^u.

Review of the Power Rule00:01:31

The power rule states that the derivative of x^n is n*x^(n-1). This is applied to understand why the derivative of 2x is 2 and the derivative of a constant is 0.

Examples of e^u Derivatives00:05:16

Several examples are worked through, including e^(3x), e^(5x), e^(x^2), e^(sin x), and e^(cos 2x), to demonstrate the application of the e^u derivative formula with various functions for u.

Nested Natural Logarithms00:26:21

Complex nested natural logarithms like ln(ln x) and ln(ln(ln x)) are differentiated using the chain rule and the ln u derivative formula, showing how to handle multiple layers of logarithmic functions.

Quotient Rule with Logarithms00:29:27

The quotient rule (g f' - f g') / g^2 is employed to find the derivative of ln x divided by x.

Logarithmic Differentiation: Variable Raised to a Variable (x^x)00:31:00

For functions like x^x (a variable raised to a variable), logarithmic differentiation is necessary. This involves taking the natural logarithm of both sides, using log properties to bring down the exponent, and then implicitly differentiating.

Shortcut Formula for Logarithmic Differentiation00:36:00

A general formula is provided for the derivative of f(x)^g(x) to streamline logarithmic differentiation. This is applied to re-solve x^x and then used for a new example: x^(sin x).

Logarithmic Differentiation for x^(ln x)00:40:30

The final example demonstrates logarithmic differentiation for x^(ln x), showing the step-by-step process of taking natural logs, using product rule, and isolating the derivative.