Use logarithmic differentiation to find the derivatives of each of the functions. f(x)=(e^2 x(x^

Use logarithmic differentiation to find the derivatives of...

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or have variables in both the base and the exponent. The method involves taking the natural logarithm of both sides of the equation, using logarithmic properties to simplify the expression, and then differentiating implicitly. This approach is particularly useful when dealing with complex expressions that would be cumbersome to differentiate using standard rules directly.

Properties of Logarithms

The properties of logarithms, such as the product rule (ln(ab) = ln(a) + ln(b)), the quotient rule (ln(a/b) = ln(a) - ln(b)), and the power rule (ln(a^c) = c ln(a)), are essential tools in logarithmic differentiation. These properties allow one to break down a complex function into simpler parts, which can then be differentiated more easily.

Chain Rule

The chain rule is a fundamental rule in calculus used to differentiate composite functions. When applying logarithmic differentiation, the chain rule often comes into play as you differentiate terms that involve a function inside another function. It ensures that the derivative of the outer function is multiplied by the derivative of the inner function, maintaining the correct overall derivative.

Exponential Function Differentiation

Differentiating exponential functions involves using specific rules, such as the fact that the derivative of e^(u(x)) is e^(u(x)) multiplied by the derivative of u(x). Exponential functions commonly appear in complex expressions, and their differentiation is a critical component when they are part of the function being analyzed.

Differentiation of Polynomial Functions

Polynomial functions are among the simplest forms of functions in calculus, and their differentiation follows the power rule, where the derivative of x^n is n*x^(n-1). In more complex expressions, polynomials often appear alongside other types of functions, and their straightforward differentiation helps simplify the overall process when using techniques like logarithmic differentiation.

Transcript

00:01 For this question here we need to recall some of the properties of derogethnic functions.

00:06 So the first one is that if we have a log of base a to two functions x and i multiply by h.

00:12 So we can separate these by saying it's equal to log of a of x plus log of a of y.

00:20 And for the case where we have a fraction x divided by y, you can separate them as well, but with a negative sign.

00:27 And finally, if we have a log of x raised to the power of n and it's just an integer, we can bring that in down next to our logarithm functions.

00:39 So now let's move on to our problem.

00:41 I have a function f of x given here.

00:44 And if we take the length of both sides, on the left hand side, we just have the line of f of x.

00:51 And here we have the length of that fraction, so we can separate them as follows.

00:58 The len.

00:59 Of e .4 2x times x per 3 minus 2 .4 minus the length of x times 3e 4x plus 1.

01:18 We can simplify this further.

01:22 The left hand side stays the same.

01:25 And here is according to the properties as well, we can separate these two functions multiplied by cheser with positive signs.

01:35 Here so we have the length e bar 2x plus the length of x.

01:44 4.

01:47 Minus here the length of x minus again the length of x minus again the length of 5x plus 1...

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