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Given that $y = 4x^3 - \frac{5}{x^2}, x \neq 0$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \, dx$ - Edexcel - A-Level Maths Pure - Question 6 - 2015 - Paper 1
Question 6
Given that $y = 4x^3 - \frac{5}{x^2}, x \neq 0$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \, dx$
Worked Solution & Example Answer:Given that $y = 4x^3 - \frac{5}{x^2}, x \neq 0$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \, dx$ - Edexcel - A-Level Maths Pure - Question 6 - 2015 - Paper 1
Step 1
(a) $\frac{dy}{dx}$
Answer
To find the derivative of the function, we first differentiate each term using the power rule.
The function is given as:
Let us rewrite the term (- \frac{5}{x^2}) as (-5x^{-2}).
Now differentiating:
Using the power rule, we have:
- The derivative of (4x^3) is (12x^2)
- The derivative of (-5x^{-2}) is (10x^{-3}) (applying the rule (nx^{n-1})).
Putting it all together:
Thus, the result is:
Step 2
(b) $\int y \, dx$
Answer
To calculate the integral of the function , we will integrate term by term. We have:
This separates into:
Now, integrating each term:
- The integral of (4x^3) is (4 \cdot \frac{x^4}{4} = x^4).
- The integral of (-5x^{-2}) is (-5 \cdot (-x^{-1}) = 5x^{-1} = \frac{5}{x}).
Combining these results gives us:
Where C is the constant of integration.