Step by step
The question
Question: find y, if dy/dx = 10x⁴ and y = 3 when x = 1.
dydx = 10x4
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
y = 10 × x55 + C
Now we have y = 2x⁵ + C.
= 2x5 + C
Every antiderivative has + C.
The point will tell us what C is.
+ C: we do not know it yet
2 Find C
Put in x = 1 and y = 3.
3 = 2(1)5 + C
Work it out: 2 + C.
= 2 + C
It must equal 3.
2 + C = 3
Take away 2 on both sides: C = 1.
C = 3 − 2 = 1
Careful!
C is not just the y value.
The other terms at x = 1 are not 0 here.
Careful: not C = 3 ✗
3 Write the function
So y = 2x⁵ + 1.
y = 2x5 + 1
4 Check
Let's check: its derivative is 10x⁴, the slope we were given.
dy/dx = 10x4
And at x = 1 it gives 3.
Right!
x = 1: 2 + 1 = 3
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
y = 2x5 + 1