Differentiation Formulas - Notes

The Organic Chemistry Tutor
27 Mar 202313:50
EducationalLearning
32 Likes 10 Comments

TLDRThis video tutorial covers essential differentiation formulas for calculus, including the derivative of constants, power rule, and chain rule. It explains how to differentiate functions involving variables raised to constants, constants raised to variables, and composite functions. The video also delves into logarithmic, trigonometric, and inverse trigonometric functions, providing formulas and examples for each. It encourages viewers to practice with additional problems and offers a resource for logarithmic differentiation.

Takeaways
  • 📝 The derivative of a constant is always zero.
  • 🔢 The power rule for differentiation states that the derivative of x^n is n*x^(n-1).
  • 📈 The derivative of a constant raised to a variable (a^x) is a^x * ln(a).
  • 🔀 For a variable raised to a variable, logarithmic differentiation is used.
  • ✖️ The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function.
  • ✏️ The product rule for differentiation is u'v + uv'.
  • ➗ The quotient rule is (vu' - uv') / v^2.
  • 🔗 The chain rule states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x).
  • 🧮 The derivative of logarithmic functions: the derivative of log base a of u is u' / (u * ln(a)) and the derivative of ln(u) is u' / u.
  • 📐 The derivatives of trigonometric functions include formulas like the derivative of sin(u) is cos(u) * u' and similar formulas for other trigonometric functions.
Q & A

  • What is the derivative of a constant?

    -The derivative of a constant is always zero.

  • What is the power rule for differentiation?

    -The power rule states that the derivative of a variable raised to a constant power N is the constant multiplied by the variable raised to the power of N minus 1, i.e., the derivative of x^N is N*x^(N-1).

  • What is the derivative of x^3 using the power rule?

    -The derivative of x^3 using the power rule is 3*x^2.

  • How do you differentiate a constant raised to a variable?

    -The derivative of a constant 'a' raised to the variable 'x' is 'a' to the power of 'x' times the natural logarithm of 'a', i.e., a^x * ln(a).

  • What is the derivative of a function multiplied by a constant?

    -The derivative of a function multiplied by a constant C is simply the constant times the derivative of the function.

  • What is the product rule for differentiation?

    -The product rule states that the derivative of the product of two functions u and v is the derivative of u times v plus u times the derivative of v, i.e., u'v + uv'.

  • What is the quotient rule for differentiation?

    -The quotient rule states that the derivative of a fraction of two functions u and v is (v * u' - u * v') / v^2.

  • How do you differentiate a composite function using the chain rule?

    -To differentiate a composite function, first find the derivative of the outer function and keep the inside part the same, then multiply by the derivative of the inside part, i.e., d/dx[f(g(x))] = f'(g(x)) * g'(x).

  • What is the derivative of a function raised to a power n?

    -The derivative of a function f(x) raised to the power n is n * f(x)^(n-1) * f'(x), combining the power rule with the chain rule.

  • What is the derivative of a logarithmic function log_a(U) where U is a function of x?

    -The derivative of log_a(U) where U is a function of x is U' / (U * ln(a)), where U' is the derivative of U with respect to x.

  • How do you differentiate the natural logarithm of a function U?

    -The derivative of the natural logarithm of U, ln(U), is U' / U, since the base of natural logarithm is e and ln(e) equals 1.

  • What is the derivative of the sine of a function U?

    -The derivative of the sine of a function U is the cosine of U times the derivative of U, i.e., cos(U) * U'.

  • What is the derivative of the inverse sine function of U?

    -The derivative of the inverse sine function of U, sin^(-1)(U), is U' / sqrt(1 - U^2).

  • What are the derivatives of the inverse trigonometric functions involving U?

    -The derivatives of the inverse trigonometric functions involving U are as follows: for arcsine, it's U' / sqrt(1 - U^2); for arccosine, it's -U' / sqrt(1 - U^2); for arctangent, it's U' / (1 + U^2); for arccotangent, it's -U' / (1 + U^2); for arcsecant, it's U' / (U * sqrt(U^2 - 1)); and for arccosecant, it's -U' / (U * sqrt(U^2 - 1)).

Outlines
00:00
📚 Introduction to Derivative Formulas in Calculus

This paragraph introduces the topic of the video, which is the explanation of various derivative formulas useful for studying derivatives in calculus. The speaker encourages viewers to have a sheet of paper ready for note-taking. The paragraph begins with the derivative of a constant, which is always zero, and proceeds to explain the power rule for derivatives of power functions, providing examples such as the derivatives of x cubed, x to the fourth, and x to the fifth. It also touches on the derivative of a constant raised to a variable and introduces logarithmic differentiation, directing viewers to a specific video on YouTube for more information.

05:00
🔍 Derivative Rules: Power, Product, Quotient, and Chain

The second paragraph delves into several key derivative rules. It starts with the constant multiple rule, explaining how to find the derivative of a function multiplied by a constant. It then revisits the power rule in the context of two functions being multiplied, leading to the product rule formula. The quotient rule is next, detailing how to differentiate a fraction of two functions. The paragraph also introduces the chain rule for composite functions, explaining the process of differentiating an outer function while keeping the inner function constant, and multiplying by the derivative of the inner function. Additional resources are mentioned in the description section for those seeking more examples.

10:06
🔄 Advanced Derivative Techniques: Chain and Power Rules Combined

This paragraph discusses advanced applications of the chain rule, particularly when combined with the power rule. It explains how to differentiate a function that is raised to a power, emphasizing the process of focusing on the outer function while keeping the inner function constant and then multiplying by the derivative of the inner function. The paragraph also presents a general form of the chain rule, expressed as dy/dx = dy/du * du/dx, and moves on to discuss the derivatives of logarithmic functions, providing formulas for both log base a of U and the natural log of U, where U is a function of x.

📉 Derivatives of Trigonometric and Inverse Trigonometric Functions

The final paragraph of the script covers the derivatives of trigonometric functions and their inverses. It starts with the derivatives of sine and cosine functions, emphasizing the use of the chain rule when the angle is a function of x. The paragraph then discusses the derivatives of tangent, cotangent, secant, and cosecant, noting the presence of negative signs and the use of secant squared or cosecant squared in their formulas. It concludes with the derivatives of inverse trigonometric functions, such as arcsine, arccosine, arctangent, and their reciprocals, providing formulas that include the use of U prime and square root terms. The video ends with a reminder of the importance of these formulas for tests on derivatives.

Mindmap
Keywords
💡Derivative
A derivative in calculus is a measure of how a function changes as its input changes. It is the slope of the tangent line to the function's graph at a particular point. The video focuses on teaching various formulas to calculate derivatives, which is central to understanding rates of change in mathematical functions.
💡Power Rule
The power rule is a fundamental principle in calculus for finding the derivative of a function where a variable is raised to a constant power. It states that the derivative of x^n (where n is a constant) is n*x^(n-1). The script uses this rule to demonstrate how to find derivatives of functions like x^3 and x^4.
💡Exponential Function
An exponential function is a mathematical function where the variable is in the exponent. The script mentions the derivative of 'a' to the power of 'x', which involves the natural logarithm of the base 'a'. This is a key concept when dealing with functions that grow or decay at a rate proportional to their current value.
💡Logarithmic Differentiation
Logarithmic differentiation is a technique used when the function to be differentiated is a variable raised to another variable. The video suggests a separate video for this topic, indicating its complexity and importance in certain differentiation problems.
💡Constant Multiple Rule
The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function. The script illustrates this with the derivative of 5x^4, which simplifies to 20x^3 using the power rule and the constant multiple rule.
💡Product Rule
The product rule is used to find the derivative of a product of two functions. The rule states that the derivative of u*v is u'*v + u*v'. The video uses this rule to explain how to differentiate the product of two functions, which is a common operation in calculus.
💡Quotient Rule
The quotient rule is a formula used to find the derivative of a quotient of two functions. It is expressed as (v*u' - u*v') / v^2. The script mentions this rule in the context of differentiating a fraction where both the numerator and the denominator are functions of x.
💡Chain Rule
The chain rule is a fundamental principle in calculus for differentiating composite functions. It involves differentiating the outer function and then multiplying by the derivative of the inner function. The video explains this rule with examples like f(g(x)) and emphasizes its importance in complex function differentiation.
💡Logarithmic Functions
Logarithmic functions are mathematical functions that are the inverse of exponential functions. The script explains how to find the derivative of a logarithmic function, such as log_a(U), which is U' / (U * ln(a)). This is crucial for understanding growth and decay processes that are not linear.
💡Trigonometric Functions
Trigonometric functions are functions of an angle, including sine, cosine, and tangent. The video discusses the derivatives of these functions, such as the derivative of sin(U) being cos(U)*U', which is essential for understanding periodic phenomena in mathematics.
💡Inverse Trigonometric Functions
Inverse trigonometric functions are the inverses of the basic trigonometric functions. The script provides formulas for their derivatives, such as the derivative of arcsin(U) being U' / sqrt(1 - U^2), which is important for solving equations involving angles in a non-linear context.
Highlights

The derivative of a constant is always zero.

Introduction of the power rule for derivatives of power functions.

Derivative of x cubed is 3x squared, demonstrating the power rule.

Explanation of the derivative of a constant raised to a variable.

Differentiation of a to the power of x involves multiplying by the natural log of a.

The constant multiple rule simplifies finding derivatives of functions multiplied by a constant.

Illustration of the power rule for the derivative of two functions multiplied together.

The quotient rule formula for the derivative of two functions divided.

The chain rule for finding derivatives of composite functions.

Combining the chain rule with the power rule for functions raised to a power.

Derivative of logarithmic functions with respect to a variable function.

Derivative formulas for natural logarithm functions.

Trigonometric derivatives: sine, cosine, tangent, and their respective rules.

Derivative of secant and cosecant functions, including the chain rule.

Inverse trigonometric functions derivatives, including arcsine and arccosine.

Derivative formulas for inverse tangent, arc cotangent, inverse secant, and inverse cosecant.

Encouragement to write down the formulas for studying derivatives.

Offering additional example problems in the description section for further understanding.

Transcripts

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DerivativesCalculusDifferentiationPower RuleLogarithmicChain RuleTrigonometryInverse TrigQuotient RuleProduct Rule