Finding the gradient to the curve using differentiation

In summary, gradient is a mathematical concept that describes the steepness or slope of a curve at a given point. It can be calculated using differentiation, which involves finding the derivative of the function at that point. Gradient and slope are closely related, with gradient representing the slope of a curve and slope describing the steepness of a straight line. In calculus, finding the gradient is important for analyzing the rate of change of a function and determining its maximum and minimum points. The gradient of a curve can be negative, positive, or zero, indicating whether the curve is decreasing, increasing, or not changing at a specific point.
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AN630078
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Homework Statement
Hello, I have been revising differentiation and calculus problems but I am rather struggling with the problem below;

The curve y=x^3-x+1 passes through the points P and Q, with x-coordinates of 1 and 1+h respectively.Using differentiation from first principles frind the gradient of the curve at P.
Relevant Equations
f'(x)=f(x+h)-f(x)/h
I have attached a photograph of my workings. I do not know if I have arrived at the right solution, nor whether this is the gradient of f(x) at point P.
I think I seem to overcomplicate these problems when thinking about them which makes me lose confidence in my answers. Thank you to anyone who replies 👍😁
 

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  • #2
Looks good to me.
 
  • #3
PeroK said:
Looks good to me.
Thank you for your reply, really splendid. I have really been trying to improve my understanding problems encompassing differentiation from first principles. 😁👍
 
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Likes berkeman

1. What is the gradient of a curve?

The gradient of a curve is a measure of how steep the curve is at a particular point. It tells us the rate of change of the curve at that point.

2. Why is differentiation used to find the gradient of a curve?

Differentiation is used to find the gradient of a curve because it is a mathematical technique that allows us to calculate the rate of change of a function at a specific point. In other words, it helps us find the slope of a curve at a particular point.

3. How do you find the gradient of a curve using differentiation?

To find the gradient of a curve using differentiation, you first need to find the derivative of the function representing the curve. The derivative is essentially the slope of the tangent line to the curve at a particular point. Once you have the derivative, you can plug in the x-value of the point you want to find the gradient at, and the resulting value will be the gradient.

4. Can you use differentiation to find the gradient at any point on a curve?

Yes, differentiation can be used to find the gradient at any point on a curve, as long as the function representing the curve is differentiable at that point. This means that the function must have a well-defined derivative at that point.

5. Are there any other methods for finding the gradient of a curve?

Yes, there are other methods for finding the gradient of a curve, such as using the slope formula or graphical methods. However, differentiation is the most commonly used method as it is more accurate and can be applied to a wider range of functions.

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