Sure, let's determine the gradient of the curve given by the equation [tex]\( y = 3x^2 - 7x + 2 \)[/tex].
Step-by-Step Solution:
1. Identify the function [tex]\( y \)[/tex]:
The given function is [tex]\( y = 3x^2 - 7x + 2 \)[/tex].
2. Understand the concept of gradient:
The gradient of a curve at any point is given by the derivative of the function with respect to [tex]\( x \)[/tex]. This derivative gives us the slope of the tangent to the curve at any point [tex]\( x \)[/tex].
3. Find the first derivative of [tex]\( y \)[/tex]:
To find the gradient, we need to take the derivative of the function [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
[tex]\[
\frac{dy}{dx} = \frac{d}{dx}(3x^2 - 7x + 2)
\][/tex]
4. Compute the derivative term-by-term:
- The derivative of [tex]\( 3x^2 \)[/tex] is [tex]\( 6x \)[/tex] (using [tex]\( \frac{d}{dx}(x^n) = nx^{n-1} \)[/tex]).
- The derivative of [tex]\( -7x \)[/tex] is [tex]\( -7 \)[/tex] (using the derivative of [tex]\( x \)[/tex]).
- The derivative of the constant [tex]\( 2 \)[/tex] is [tex]\( 0 \)[/tex] (since the derivative of a constant is zero).
5. Combine the results:
[tex]\[
\frac{dy}{dx} = 6x - 7
\][/tex]
6. Conclusion:
The gradient of the curve [tex]\( y = 3x^2 - 7x + 2 \)[/tex] at any point [tex]\( x \)[/tex] is given by [tex]\( 6x - 7 \)[/tex].
So, the gradient (or derivative) of the curve is [tex]\( 6x - 7 \)[/tex].
