# 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

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