Photo AI
Given that $y = 2x^4 - \frac{6}{x^2}$, where $x \neq 0$ - Edexcel - A-Level Maths Pure - Question 6 - 2006 - Paper 1
Question 6
Given that $y = 2x^4 - \frac{6}{x^2}$, where $x \neq 0$. (a) find \( \frac{dy}{dx} \). (b) find \( \int y \; dx \).
Worked Solution & Example Answer:Given that $y = 2x^4 - \frac{6}{x^2}$, where $x \neq 0$ - Edexcel - A-Level Maths Pure - Question 6 - 2006 - Paper 1
Step 1
find \( \frac{dy}{dx} \)
Answer
To find the derivative of ( y ) with respect to ( x ), we will apply the power rule of differentiation:
- Differentiate the first term: ( 2x^4 ) gives ( \frac{d}{dx}(2x^4) = 8x^3 ).
- Differentiate the second term: ( -\frac{6}{x^2} ) can be rewritten as ( -6x^{-2} ), leading to ( \frac{d}{dx}(-6x^{-2}) = 12x^{-3}. )
Thus, combining these gives:
Step 2
find \( \int y \; dx \)
Answer
To find the integral of ( y ) with respect to ( x ), we will integrate each term:
- The integral of ( 2x^4 ) is ( \int 2x^4 , dx = \frac{2}{5}x^5 + C_1 ).
- The integral of ( -\frac{6}{x^2} ) is ( \int -6x^{-2} , dx = 6x^{-1} + C_2. )
Putting these together, we have:
where ( C ) is the constant of integration.