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Question 1

$y = 2x^3 + 5x^2 - 7x + 10$
(a) Find $\frac{dy}{dx}$.
(b) Find the gradient of the curve when $x = 2$.

Solution for Question 1:

(a) Differentiate $y$: $\frac{dy}{dx} = 6x^2 + 10x - 7$

(b) Substitute $x = 2$ into $\frac{dy}{dx}$: $\frac{dy}{dx} = 6(2)^2 + 10(2) - 7 = 24 + 20 - 7 = 37$

The gradient of the curve when $x = 2$ is 37.

Would you like me to proceed with the next question or provide more details?
Here are 5 questions based on this explanation:

1. What is the rule for differentiating $ax^n$?
2. How do we evaluate the gradient at a specific $x$-value?
3. What is the significance of $\frac{dy}{dx}$ in calculus?
4. How can you use the derivative to check if a function is increasing or decreasing?
5. How would the derivative change if the function included trigonometric terms?

Tip: Always simplify expressions after differentiating before substituting values.