Which Derivative Is Described By The Following Expression

Let's delve into the fascinating world of derivatives and explore how to identify the derivative being described by a given mathematical expression. Understanding this involves recognizing common derivative rules, applying the chain rule appropriately, and interpreting the notation used. This knowledge is crucial for solving problems in calculus, physics, engineering, economics, and numerous other fields.

Fundamentals of Derivatives

Before dissecting complex expressions, it's important to revisit the foundational concepts. The derivative of a function, denoted as dy/dx (Leibniz notation) or f'(x) (Lagrange notation), represents the instantaneous rate of change of the function y with respect to the variable x. Geometrically, it's the slope of the tangent line to the function's graph at a specific point.

Here's a quick rundown of some fundamental derivative rules:

• Power Rule: If y = xⁿ, then dy/dx = nx^n-1.
• Constant Rule: If y = c (where c is a constant), then dy/dx = 0.
• Constant Multiple Rule: If y = cf(x), then dy/dx = cf'(x).
• Sum/Difference Rule: If y = u(x) ± v(x), then dy/dx = u'(x) ± v'(x).
• Product Rule: If y = u(x)v(x), then dy/dx = u'(x)v(x) + u(x)v'(x).
• Quotient Rule: If y = u(x)/v(x), then dy/dx = [v(x)u'(x) - u(x)v'(x)] / [v(x)]².
• Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
• Derivatives of Trigonometric Functions:
  • d/dx (sin x) = cos x
  • d/dx (cos x) = -sin x
  • d/dx (tan x) = sec² x
  • d/dx (csc x) = -csc x cot x
  • d/dx (sec x) = sec x tan x
  • d/dx (cot x) = -csc² x
• Derivatives of Exponential and Logarithmic Functions:
  • d/dx (e^x) = e^x
  • d/dx (a^x) = a^x ln(a)
  • d/dx (ln x) = 1/x
  • d/dx (log_a x) = 1 / (x ln(a))

Decoding Derivative Expressions: A Step-by-Step Approach

Let's outline a systematic approach to deciphering which derivative a given expression represents.

1. Identify the Variables: Determine the independent and dependent variables involved. This is crucial for understanding the context of the derivative. For example, in dy/dt, y is the dependent variable and t is the independent variable, implying that y is a function of t.

2. Recognize the Notation: Pay close attention to the notation used:

   • dy/dx, dz/dt, df/du: Leibniz notation, indicating the derivative of y with respect to x, z with respect to t, and f with respect to u, respectively.
   • f'(x), g'(t), h'(z): Lagrange notation, representing the first derivative of the function f, g, and h with respect to their respective variables.
   • f''(x), y'', d²y/dx²: Notation for the second derivative. The second derivative represents the rate of change of the first derivative, and it's often interpreted as the concavity of the function.
   • ∂f/∂x, ∂z/∂y: Partial derivative notation, used for multivariable functions. This signifies the derivative of f with respect to x while holding other variables constant.

3. Look for Derivative Rules in Action: Analyze the structure of the expression to identify which derivative rules might have been applied. For example, if you see terms being added or subtracted, the sum/difference rule likely played a role. If you see a product of two functions, the product rule is probably involved.

4. Unravel the Chain Rule: The chain rule is often a source of complexity. If you encounter a function within a function (a composite function), the chain rule is almost certainly in play. Identify the "outer" and "inner" functions and how their derivatives are related.

5. Simplify and Interpret: Simplify the expression as much as possible. This will often reveal the underlying function and the derivative rule(s) that were used to obtain the expression. Interpret the result in the context of the problem. What does the derivative tell you about the function's behavior?

Illustrative Examples

Let's work through several examples to solidify these concepts:

Example 1: Expression: 2x

• Variables: Implied y is a function of x. We are looking for dy/dx.
• Notation: The expression represents a derivative.
• Derivative Rule: This looks like the result of applying the power rule. If y = x², then dy/dx = 2x.
• Interpretation: The expression 2x represents the derivative of x² with respect to x.

Example 2: Expression: cos(x)

• Variables: Implied y is a function of x.
• Notation: The expression represents a derivative.
• Derivative Rule: This is the direct derivative of the sine function. If y = sin(x), then dy/dx = cos(x).
• Interpretation: The expression cos(x) represents the derivative of sin(x) with respect to x.

Example 3: Expression: e^x

• Variables: Implied y is a function of x.
• Notation: The expression represents a derivative.
• Derivative Rule: This is the derivative of the exponential function e^x. If y = e^x, then dy/dx = e^x.
• Interpretation: The expression e^x represents the derivative of e^x with respect to x.

Example 4: Expression: 2x cos(x²)

• Variables: Implied y is a function of x.
• Notation: The expression represents a derivative.
• Derivative Rule: This looks like a combination of the chain rule and the derivative of sine.
• Chain Rule Unraveling: If y = sin(x²), then dy/dx = cos(x²) * (2x), which simplifies to 2x cos(x²).
• Interpretation: The expression 2x cos(x²) represents the derivative of sin(x²) with respect to x.

Example 5: Expression: (x cos(x) + sin(x))

• Variables: Implied y is a function of x.
• Notation: The expression represents a derivative.
• Derivative Rule: This suggests the product rule might be involved.
• Product Rule Unraveling: Consider y = x sin(x). Using the product rule, dy/dx = (1)sin(x) + x(cos(x)) = sin(x) + x cos(x).
• Interpretation: The expression (x cos(x) + sin(x)) represents the derivative of x sin(x) with respect to x.

Example 6: Expression: -2e^-2x

• Variables: Implied y is a function of x.
• Notation: The expression represents a derivative.
• Derivative Rule: This involves the chain rule and the derivative of the exponential function.
• Chain Rule Unraveling: If y = e^-2x, then dy/dx = e^-2x * (-2) = -2e^-2x.
• Interpretation: The expression -2e^-2x represents the derivative of e^-2x with respect to x.

Example 7: Expression: 1/(2√x)

• Variables: Implied y is a function of x.
• Notation: The expression represents a derivative.
• Derivative Rule: This looks like the power rule applied to a fractional exponent.
• Power Rule Unraveling: If y = √x = x^1/2, then dy/dx = (1/2)x^{(1/2 - 1)} = (1/2)x^-1/2 = 1/(2√x).
• Interpretation: The expression 1/(2√x) represents the derivative of √x with respect to x.

Example 8: Expression: (2x(x² + 1)⁴)

• Variables: Implied y is a function of x.
• Notation: The expression represents a derivative.
• Derivative Rule: This involves the chain rule and the power rule. Let's try to reverse engineer it. It likely comes from something like (x² + 1)⁵.
• Chain Rule Unraveling: If y = (1/5)(x² + 1)⁵, then dy/dx = (1/5) * 5(x² + 1)⁴ * (2x) = 2x(x² + 1)⁴. Notice the constant 1/5. This is because the original function could have any constant multiplied in front of it, and the derivative expression will be the same.
• Interpretation: The expression 2x(x² + 1)⁴ represents the derivative of (1/5)(x² + 1)⁵ with respect to x.

Example 9: Expression: sec²(x)

• Variables: Implied y is a function of x.
• Notation: The expression represents a derivative.
• Derivative Rule: This is the derivative of the tangent function.
• Interpretation: The expression sec²(x) represents the derivative of tan(x) with respect to x.

Example 10: Expression: -sin(x)e^cos(x)

• Variables: Implied y is a function of x.
• Notation: The expression represents a derivative.
• Derivative Rule: Chain rule is definitely involved, likely with exponential and trigonometric functions.
• Chain Rule Unraveling: Let's consider y = e^cos(x). Then, dy/dx = e^cos(x) * (-sin(x)) = -sin(x)e^cos(x).
• Interpretation: The expression -sin(x)e^cos(x) represents the derivative of e^cos(x) with respect to x.

Dealing with Higher-Order Derivatives

The same principles apply when dealing with higher-order derivatives (second derivative, third derivative, etc.). You simply need to recognize the notation (e.g., f''(x), d²y/dx²) and apply the derivative rules repeatedly.

Example: Expression: 6x

• We know from Example 1 that 2x is the derivative of x². Therefore, 6x is the derivative of 3x².
• Now, what function has a first derivative of 3x²? That would be x³.
• Therefore, 6x can be interpreted as the second derivative of x³. We could write f''(x) = 6x if f(x) = x³.

Partial Derivatives: Extending to Multivariable Functions

When dealing with functions of multiple variables, we use partial derivatives. The partial derivative of a function f(x, y) with respect to x, denoted as ∂f/∂x, represents the rate of change of f with respect to x while holding y constant. Similarly, ∂f/∂y represents the rate of change of f with respect to y while holding x constant.

Example: Expression: 2x + y

• Variables: The expression involves both x and y, suggesting a multivariable function.
• Notation: This could be a partial derivative.
• Partial Derivative Unraveling: Let's consider f(x, y) = x² + xy. Then, ∂f/∂x = 2x + y.
• Interpretation: The expression 2x + y represents the partial derivative of x² + xy with respect to x.

Common Pitfalls and How to Avoid Them

• Forgetting the Chain Rule: This is a very common error. Always double-check if you have a composite function.
• Incorrectly Applying the Product or Quotient Rule: Make sure you have the correct order of terms and signs in the product and quotient rules.
• Confusing Variables: Clearly identify the independent and dependent variables, especially in implicit differentiation problems.
• Not Simplifying: Simplifying the expression can make it much easier to recognize the original function and the derivative rule(s) that were applied.
• Ignoring Constants: Remember to account for constant multiples when reversing the derivative process.
• Assuming a Unique Solution: There might be multiple functions that could have led to the same derivative expression (differing by a constant, for example).

Practice Exercises

To further hone your skills, try identifying the original function (or a possible original function) for each of the following derivative expressions:

1. 3x² + 4x - 1
2. cos(2x)
3. x / √(x² + 1)
4. 2sec²(2x)tan(2x)
5. (1 - ln(x)) / x²
6. ∂f/∂x = 3x²y² + 2y

Conclusion

Identifying the derivative described by a given expression is a valuable skill that requires a solid understanding of derivative rules, the chain rule, and proper notation. By systematically analyzing the expression and reversing the differentiation process, you can successfully determine the original function (or a possible original function) and gain deeper insights into the relationships between functions and their derivatives. Consistent practice and attention to detail are key to mastering this skill and excelling in calculus and related fields. Remember to always double-check your work and consider the context of the problem to ensure accurate interpretation.