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“Right so at this point. You ve now seen some of the shortcut rules for for finding derivatives. Which is going to make life a lot easier at least when comes to finding derivatives. So here we just got three examples and i m just going to find the derivative of each one of these so the first one s quite easy so.
We ve got f of x. Equals. X. To the tenth.
So the derivative rule. Says that whenever you have a variable to a power. We just pull that number out front. So the 10 is going to come right out front and then we simply subtract 1 from our exponent.
So we would have 10 minus 1 or 9. And that s our derivative. It. Says hey.
The derivative is just going to be 10 times x to the 9th so you know if you tried to use that long definition f of x plus h minus f of x over h you d take you quite a bit of work to get there so kind of nice. How quick and easy we can get this solution. Let s look at our next. One.
Here g. Of. X..
Equals. X. To. The 5 halves.
Plus 3. X. Plus. 4.
Well ok. We ll do the same thing. Here ok. So the 5 over.
2. That s going to come out front. Let s write this a little better so our. 5 over 2 is going to come right out front.
Ok. Then we have to subtract 1. So maybe i ll write that so we have to do 5. Over 2 minus.
1. That s going to be our new exponent. Plus well to kind of be really clear here..
When you re multiplying by a constant nothing really happens to the constant. We can just take the derivative of the sort of the variable part so the 1 would come out front and then we would subtract 1 from the exponent and that would give us. 0. Okay so we ll come back to this in just a second the derivative of a constant is just 0.
So the plus 4. Normally we don t write plus 0. But i ll write it here just to be you know kind of hopefully clear about what s going on all right well. So we ve got five over two times x to the five halves well.
We would have to get common denominators. So instead of minus. One we could write that as two over two well five minus two will give us three over two when we take a number and raise it to the zero power. We re going to get just one so we can just write three times x.
To the zero is three times one again. The plus zero. We ll leave it off so. It says.
Our derivative is going to be five half five halves times x. Raised to the 3 over 2 plus. Three. So.
What happens is again you know the five over two just comes out front like normal you know hey it s a fraction. But if you subtract one you re just subtracting two over two in this case. Which gives us three halves..
When you have a number times. A variable to the first basically the variable just goes away. When you take the derivative and then the constant just goes to zero. So that would be our derivative for part b.
And last. But not least let s look at our last. One here so we ve got so we had h of x. Equals.
One over the cubed root of x. And maybe just to remind you remember the so. If we have x raised to the m and then the in througt of that we can write that as x raised to the m over n. So you can think about this.
As being x to the first power underneath. So we could really rewrite this as 1 over x raised to the 1 3. Power well to use that derivative rule. We need sort of the x to be in the numerator well.
I can pull the x upstairs. And i just have to change the sign on the exponent so really h of x. I can rewrite that as just saying. It s x to the negative 1 3.
And now i m in a good spot. Where i can take the derivative. So the negative 1 3..
Comes right out front and then we would have negative 1 3. And we would subtract one. But i can write 1 as 3 over 3 to get common denominators. So we would have negative.
1 over 3 well negative one minus three will give us negative four over three and this is now our derivative. It says it would be negative. One third x to the negative four thirds we could rewrite this as negative. 1 over 3x to the positive 4 3.
And if we wanted to we could even put this back in our radical notation. So kind of using this little rule up here. So it says. Whatever is in the denominator that s what the root is so here we would be taking still a cube root.
But now instead of just having x to the first we would have x to the fourth power. So that s how we could sort of rewrite it all again using radical notation. So all right again just kind of some basic rules here. You ll use these all the time in calculus.
Certainly will be some other rules coming along as well. But again certainly. I think once you ve done a few of these using the definition life seems much easier now so. But again now the fun part is they re going to give you really complicated functions to take the derivative of so it can still be a little tricky.
But certainly these shortcuts do make life a lot easier. ” ..
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