AS Pure Chapter 12: Differentiation

16 flashcards · Mathematics · by TheoTyson

1. What is the gradient of a curve at a given point?

   The gradient of the tangent to the curve at that point.

2. What is the gradient / derivative notation?

   f′(x)f'(x)f′(x) or dydx\frac{dy}{dx}dxdy​

3. What is the formula for differentiation from first principles?

   f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​

4. What is the derivative of f(x)=xnf(x) = x^nf(x)=xn ?

   f′(x)=nxn−1f'(x) = nx^{n-1}f′(x)=nxn−1

5. What is the derivative of f(x)=axnf(x) = ax^nf(x)=axn ?

   f′(x)=anxn−1f'(x) = anx^{n-1}f′(x)=anxn−1

6. What is the derivative of y=f(x)+g(x)y = f(x) + g(x)y=f(x)+g(x) ?

   dydx=f′(x)+g′(x)\frac{dy}{dx} = f'(x) + g'(x)dxdy​=f′(x)+g′(x)

7. What is the equation of the tangent to the curve y=f(x)y = f(x)y=f(x) at the point (a,f(a))(a, f(a))(a,f(a)) ?

   y−f(a)=f′(x)(x−a)y - f(a) = f'(x)(x - a)y−f(a)=f′(x)(x−a)

8. What is the equation of the normal to the curve y=f(x)y = f(x)y=f(x) at the point (a,f(a))(a, f(a))(a,f(a)) ?

   y−f(a)=−1f′(x)(x−a)y - f(a) = -\frac{1}{f'(x)}(x - a)y−f(a)=−f′(x)1​(x−a)

9. When is a function increasing?

   When f′(x)⩾0f'(x) \geqslant 0f′(x)⩾0 .

10. When is a function decreasing?

    When f′(x)⩽0f'(x) \leqslant 0f′(x)⩽0 .

11. How do you find the second derivative of y=f(x)y = f(x)y=f(x) ?

    Differentiate the function twice to get f′′(x)f''(x)f′′(x) or d2ydx2\frac{d^2 y}{d x^2}dx2d2y​ .

12. What are the gradients before and after a local maximum turning point?

    The gradient before is positive, and the gradient after is negative.

13. What are the gradients before and after a local minimum turning point?

    The gradient before is negative, and the gradient after is positive.

14. What are the gradients before and after a point of infection turning point?

    The gradient before will be the same as the gradient after the point, both positive or both negative.

15. What does it mean if f′(a)=0f'(a) = 0f′(a)=0 and f′′(a)<0f''(a) < 0f′′(a)<0 ?

    Then x=ax=ax=a is a local maximum.

16. What does it mean if f′(a)=0f'(a) = 0f′(a)=0 and 0"> f′′(x)>0f''(x) > 0f′′(x)>0 ?

    Then x=ax=ax=a is a local minimum.