Gradients and second derivatives questions with answers

Questions

1. Question 1The gradient at a point

   Find the gradient of the curve $y = 2x^{3} + 3x^{2} - 5x$ at the point where $x = 3$.

   Show the worked solution

   1. Step 1: Differentiate each term: bring the power down, take 1 off the power$\dfrac{dy}{dx} = 2(3)x^{2} + 3(2)x - 5(1)$$= 6x^{2} + 6x - 5$
   2. Step 2: Put the given x into dy/dx$\dfrac{dy}{dx} = 6(3)^{2} + 6(3) - 5$$= 67$
2. Question 2The gradient at a point

   Find the gradient of the curve $y = -x^{3} + 6x^{2} + 1$ at the point where $x = 1$.

   Show the worked solution

   1. Step 1: Differentiate each term: bring the power down, take 1 off the power$\dfrac{dy}{dx} = -3x^{2} + 6(2)x$$= -3x^{2} + 12x$
   2. Step 2: Put the given x into dy/dx$\dfrac{dy}{dx} = -3(1)^{2} + 12(1)$$= 9$
3. Question 3Second derivatives

   Given that $y = x^{5} - 10x^{2}$, find $\dfrac{d^{2}y}{dx^{2}}$.

   Show the worked solution

   1. Step 1: Differentiate each term: bring the power down, take 1 off the power$\dfrac{dy}{dx} = 5x^{4} - 10(2)x$$= 5x^{4} - 20x$
   2. Step 2: Differentiate again for the second derivative$\dfrac{d^{2}y}{dx^{2}} = 5(4)x^{3} - 20(1)$$= 20x^{3} - 20$
4. Question 4Second derivatives

   Given that $y = x^{6} + \dfrac{2}{x^{2}}$, find the value of $\dfrac{d^{2}y}{dx^{2}}$ when $x = 1$.

   Show the worked solution

   1. Step 1: Write every term as a power of x$y = x^{6} + 2x^{-2}$
   2. Step 2: Differentiate each term: bring the power down, take 1 off the power$\dfrac{dy}{dx} = 6x^{5} + 2(-2)x^{-3}$$= 6x^{5} - 4x^{-3}$
   3. Step 3: Differentiate again for the second derivative$\dfrac{d^{2}y}{dx^{2}} = 6(5)x^{4} - 4(-3)x^{-4}$$= 30x^{4} + 12x^{-4}$
   4. Step 4: Put the given x into the second derivative$\dfrac{d^{2}y}{dx^{2}} = 30(1)^{4} + 12(1)^{-4}$$= 42$
5. Question 5Where the gradient has a given value

   Find the $x$-coordinates of the points on the curve $y = x^{3} - 6x^{2} + 9x + 1$ at which the gradient is $0$.

   Show the worked solution

   1. Step 1: Differentiate each term: bring the power down, take 1 off the power$\dfrac{dy}{dx} = 3x^{2} - 6(2)x + 9(1)$$= 3x^{2} - 12x + 9$
   2. Step 2: Set dy/dx equal to the given gradient, and bring everything to one side$3x^{2} - 12x + 9 = 0$
   3. Step 3: Divide through by the common factor$x^{2} - 4x + 3 = 0$
   4. Step 4: Factorise and solve$(x - 1)(x - 3) = 0$$x = 1$ or $x = 3$
6. Question 6Where the gradient has a given value

   Find the $x$-coordinates of the points on the curve $y = x^{3} - 6x^{2} + 6x + 1$ at which the gradient is $-3$.

   Show the worked solution

   1. Step 1: Differentiate each term: bring the power down, take 1 off the power$\dfrac{dy}{dx} = 3x^{2} - 6(2)x + 6(1)$$= 3x^{2} - 12x + 6$
   2. Step 2: Set dy/dx equal to the given gradient, and bring everything to one side$3x^{2} - 12x + 9 = 0$
   3. Step 3: Divide through by the common factor$x^{2} - 4x + 3 = 0$
   4. Step 4: Factorise and solve$(x - 1)(x - 3) = 0$$x = 1$ or $x = 3$