Math Problem Statement

Yasemin tried to find the derivative of [\cos(x)\cdot x^6]. Here is her work: Step 1: This is the product of [\cos(x)] and [x^6]. So we should use the product rule. Step 2:

[\dfrac{d}{dx}[\cos(x)\cdot x^6]=\dfrac{d}{dx}[\cos(x)]x^6+\cos(x)\dfrac{d}{dx}[x^6]] Step 3: Finding the derivatives of the factors:

[\begin{aligned} \dfrac{d}{dx}[\cos(x)]&=-\sin(x) \\ \dfrac{d}{dx}[x^6]&=6x^5 \end{aligned}] Step 4: Putting it all together:

[\begin{aligned} &\phantom{{}={}}\dfrac{d}{dx}[\cos(x)\cdot x^6] \\ &=\dfrac{d}{dx}[\cos(x)]x^6+\cos(x)\dfrac{d}{dx}[x^6] \\ &=-\sin(x)\cdot x^6+\cos(x)\cdot 6x^5 \end{aligned}] Is Yasemin's work correct? If not, what's her mistake?

Solution

Yasemin's work is correct. She properly applied the product rule and correctly computed the derivatives of the individual terms.

Verifying her steps:

  1. Step 1: Recognizing that the derivative involves the product of two functions (cos⁡(x)\cos(x) and x6x^6) is correct.
  2. Step 2: Correctly writes the product rule: [ \dfrac{d}{dx}[f(x)\cdot g(x)] = \dfrac{d}{dx}[f(x)]\cdot g(x) + f(x)\cdot \dfrac{d}{dx}[g(x)] ] Here, f(x)=cos⁡(x)f(x) = \cos(x) and g(x)=x6g(x) = x^6.
  3. Step 3: Computes the derivatives:
    • ddx[cos⁡(x)]=−sin⁡(x)\dfrac{d}{dx}[\cos(x)] = -\sin(x), which is correct.
    • ddx[x6]=6x5\dfrac{d}{dx}[x^6] = 6x^5, which is also correct.
  4. Step 4: Substitutes back into the product rule: ddx[cos⁡(x)⋅x6]=−sin⁡(x)⋅x6+cos⁡(x)⋅6x5\dfrac{d}{dx}[\cos(x)\cdot x^6] = -\sin(x)\cdot x^6 + \cos(x)\cdot 6x^5 This is both correct and fully simplified.

Final Answer:

Yasemin has no mistakes in her work.

Would you like further explanation of the product rule or examples of similar problems?

Here are 5 follow-up questions to expand your understanding:

  1. What is the product rule, and why is it necessary for differentiating products of functions?
  2. How does the order of terms affect the result in the product rule?
  3. What happens when one of the functions in the product is a constant?
  4. Could we rewrite cos⁡(x)⋅x6\cos(x)\cdot x^6 before differentiating? Would it simplify the derivative?
  5. How would the derivative change if the product included a coefficient (e.g., 3cos⁡(x)⋅x63\cos(x)\cdot x^6)?

Tip: Always confirm that all intermediate derivatives are computed correctly when using the product rule to ensure the final result is accurate!

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Product Rule

Formulas

Product rule: \(\dfrac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\)
Derivative of \(\cos(x)\): \(\dfrac{d}{dx}[\cos(x)] = -\sin(x)\)
Power rule: \(\dfrac{d}{dx}[x^n] = nx^{n-1}\)

Theorems

Product Rule

Suitable Grade Level

Grades 11-12

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