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Differentiation: Exercise 2 – Calculus Methods

by Batool Akmal
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    00:01 So moving on from the previous question where we were just asked to find the general gradient.

    00:07 This questions move up one step further, it's asking you the gradient at x equals to 5.

    00:12 So all that means is that we do the same process like in the general gradient and then we substitute 5 into the equation.

    00:20 So there is no need to change any roots or any indices here.

    00:24 So we have y equals 10x to the 5 minus 2x plus 7.

    00:30 Just rewrite it again so we can see where the powers are going.

    00:33 dy by dx.

    00:37 Bring the power down.

    00:38 So we have 5 times 10x, 5 minus 1 minus 1.

    00:46 And then we've got 2x, 1 minus 1.

    00:49 And anything as a constant just disappears.

    00:52 This gives me 50x to the 4 minus 2.

    00:59 We are now looking at the gradient at x equals to 5.

    01:03 So we've got 50x to the 5.

    01:07 Let's just x to the 4 I mean.

    01:08 Let's just say what we're doing, so dy by dx at x = 5 - 2.

    01:17 You'd have to work out what 5 to the power of 4 is.

    01:21 So we know where 5 squared is, means 25.

    01:24 You multiply that with another 5 to give you a 125.

    01:29 You multiply that with another 5.

    01:31 So again remember that we are not quite using calculators here.

    01:35 So you can't just work it out on the side.

    01:37 And it's important that you do.

    01:39 Because there's no shame in working things on the side as long as you don't get it wrong.

    01:43 So you have 25.

    01:45 5 times 2 is 10 and plus 2 gives you 12.

    01:48 And then we've got 5 times 1.

    01:49 So we got 625.

    01:51 So here we can rewrite this as 50 times 600 and 25 minus 2.

    01:59 So now we're faced with 50 multiplied by 600 and 25 which looks fun.

    02:05 So let's just multiply that here.

    02:07 Any method of multiplication obviously is fine.

    02:10 And I usually do the 0 with all three numbers and then the 5.

    02:14 So that gives me 0, 0, 0.

    02:16 And then I now multiply 5 with the 5 to give me 25.

    02:20 5 times 2 is 10 plus the 2 gives me 12.

    02:24 And I got 5 times 6 is 30 plus 1, gives me 31.

    02:29 So you add them up now.

    02:31 So that gives you 31,250.

    02:35 So hopefully things won't be as complicated when you do these calculations.

    02:42 But I've got 31, 250 minus 2.

    02:46 Which then gives me 31, 248 as the gradient.

    02:51 Which is extremely steep but positive.

    02:54 So the gradient of 10x to 5 minus 2x plus 7, you have your general gradient here.

    03:02 And then your particular gradient at x = 5 is this.

    03:07 So you can imagine that this curve is very, very steep as your x values are growing.


    About the Lecture

    The lecture Differentiation: Exercise 2 – Calculus Methods by Batool Akmal is from the course Calculus Methods: Differentiation.


    Included Quiz Questions

    1. 45x⁸+5
    2. 9x⁸+5
    3. 45x⁸+5x
    4. 9x⁸+5x
    5. 45x⁸-5
    1. y=2,y=x+2,y=2x,y=x²
    2. y=x²,y=x+2,y=2x,y=2
    3. y=2,y=2x,y=x+2,y=x²
    4. y=2x,y=x²,y=x+2,y=2
    5. y=2x,y=x²,y=2,y=x+2

    Author of lecture Differentiation: Exercise 2 – Calculus Methods

     Batool Akmal

    Batool Akmal


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