Summary of [Derivatives for Beginners

Segment

00:00 - This video segment discusses how to find derivatives. It starts by explaining that the derivative of any constant is always zero. This means that the derivative of numbers like 7, -4, pi, and pi to the e is all zero. The segment then mentions that if there is no variable and only a constant, the derivative will always be zero.
00:54 - The power rule in calculus states that if a variable is raised to a constant power, the derivative is equal to the constant multiplied by the variable raised to the power minus one. For example, the derivative of x squared is 2x, the derivative of x cubed is 3x squared, the derivative of x to the fourth power is 4x cubed, and the derivative of x to the fifth power is 5x to the fourth power. This pattern can be applied to find the derivative of any variable raised to a constant power.
02:08 - The video segment discusses the concept of finding the derivative of a function with a constant multiple. It explains that to find the derivative of a constant multiplied by a function, you can simply multiply the constant by the derivative of the function. This is known as the constant multiple rule. The segment provides two examples to illustrate this rule: the derivative of 6x^8 is 48x^7, and the derivative of 5x^3 is 15x^2.
03:51 - The video segment explains how to find the derivative of a polynomial function step by step. It demonstrates the process using the example of a polynomial function, f(x) = 4x^3 + 7x^2 - 9x + 5. The derivative of each term is calculated using the power rule, where the exponent is multiplied by the coefficient and then reduced by one. The final result is the derivative of the original function, which in this case is 12x^2 + 14x - 9.
06:05 - The video segment explains how to find the derivative of rational functions using the power rule. The first step is to rewrite the expression by moving the variable to the top and changing the sign of the exponent. Then, using the power rule, the exponent is moved to the front and subtracted by one. Finally, the variable is moved back to the bottom to simplify the expression. The segment provides examples of finding the derivatives of 1/x^2, 1/x^3, and -6/x^5, demonstrating the step-by-step process.
09:37 - The video segment explains how to find the derivative of radical functions. The first example demonstrates finding the derivative of the square root of x, which simplifies to 1/2 times the square root of x. The second example involves finding the derivative of the cube root of x to the fifth power, which simplifies to 5 times x raised to the 2/3 power. The final example involves finding the derivative of the eighth root of x to the fifth power, which simplifies to 5 over 8 times x raised to the 3/8 power. The segment emphasizes the steps of rewriting the expression, moving the exponent to the front, and simplifying the exponent.
14:38 - The video segment discusses the derivatives of trigonometric functions. It explains the derivatives of sine, cosine, tangent, cotangent, secant, and cosecant functions, highlighting the patterns and similarities between them. The segment then provides examples of finding the derivatives of various trigonometric functions, such as sine x and sine x cubed, cosine x squared, tangent x to the fifth power, secant 4x, and cotangent x cubed plus x to the fifth power. The process involves differentiating the trigonometric function and the inside part separately, while keeping the angle the same in the answer. The segment concludes by providing the derivatives for each example.
21:35 - The video segment discusses the derivatives of natural logarithms. It explains that the derivative of ln u is equal to u prime divided by u. The segment provides examples of finding the derivatives of ln x and ln x cubed, demonstrating two different methods. It also shows how to find the derivative of ln x to the fourth minus x to the fifth. Additionally, the segment explains how to find the derivative of ln tangent x and simplifies the expression to cosecant x times secant x.
25:47 - The video segment discusses how to find the derivative of a regular logarithmic function, such as log base a of u. The derivative is found using the formula u prime over u ln a. The video compares this formula to the derivative of the natural log of u, which is u prime over u ln e. It highlights the similarities between the two equations and demonstrates how to find the derivative of specific logarithmic functions, such as log base 2 of x to the fifth power and log base 4 of x cubed plus 4x squared. The segment concludes by simplifying the answers and providing different ways to write the final answer.
30:33 - The video segment discusses the derivative of exponential functions. It explains that when the base is e, the derivative is simply the exponential function multiplied by the derivative of the variable. However, if the base is a constant other than e, the derivative also includes the natural logarithm of the base. The segment provides examples of finding the derivatives of e to the x, e to the 2x, e to the 5x, e to the x squared, and e to the sine x. It then demonstrates finding the derivatives of 5 raised to the x and 7 raised to the x to the fourth, highlighting the use of the natural logarithm of the base in these cases.
35:07 - The video segment discusses the product rule in calculus. It explains that when finding the derivative of a function multiplied by another function, you differentiate the first part and leave the second part the same, then leave the first part the same and differentiate the second part. The segment provides examples of applying the product rule to different functions and simplifying the derivatives. It also mentions an alternative method of finding the derivative by distributing before taking the derivative.
40:09 - The video segment explains how to use the quotient rule to find the derivative of a function divided by another function. The formula for the quotient rule is v u prime minus u v prime over v squared. The segment provides an example of finding the derivative of (3x - 5) divided by (7x + 4) using the quotient rule. By plugging in the values and simplifying the expression, the final answer is determined to be 47 divided by (7x + 4) squared. The video concludes by stating that the quotient rule can be used whenever there is a division of two functions.
42:38 - The video segment discusses the chain rule in calculus. It explains that when dealing with composite functions, the derivative of the outer function is multiplied by the derivative of the inner function. The segment provides examples of applying the chain rule to find derivatives of functions with composite functions. It emphasizes the importance of differentiating each function step by step, starting from the outermost function and working inward. The segment also highlights the use of the power rule when dealing with exponents in composite functions.
48:27 - Implicit differentiation is used when you have equations with two different variables and want to find the derivative of one variable with respect to the other. To do this, you differentiate both sides of the equation with respect to the desired variable. When differentiating x cubed with respect to x, you get 3x squared, but when differentiating y cubed with respect to x, you get 3y squared times dy/dx. In related rates problems, you differentiate with respect to a different variable, such as time. In the example given, the derivative of x to the fourth is 4x cubed, but the derivative of y to the fourth is 4y cubed times dy/dx. To solve for dy/dx, you isolate the term and divide both sides by the appropriate factor. Another example is provided to further illustrate the process.
53:49 - The video segment explains how to find the derivative of a variable raised to a variable using logarithmic differentiation. The process involves setting y equal to x raised to the x, taking the natural log of both sides, differentiating both sides with respect to x, and using the product rule. The final answer is x raised to the x times 1 plus ln x. This method is known as logarithmic differentiation and allows for finding the derivative of a variable raised to a variable.

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