Derivative of a Linear Combination: Overview
Hi, everyone.
Welcome back to integralcalc.com.
Today we’re going to be talking about how to find the derivative of a linear combination.
And it sounds really complicated but it’s not.
Basically, the formula that we’re going to be using here is the following.
All it says is that when you’re taking the derivative of some function, f of x,
and it’s multiplied by a constant real number a like this.
You’re taking the derivative of that whole thing. You can just factor out the a.
You bring it out in front here and you can ignore that for the purposes of the derivative
and just take the derivative of the function f of x and multiplying them together,
a times f of x will be the final answer for the derivative. So pretty easy.
Let’s just look at a couple examples.
We’ve got five x to the fourth. What it’s saying here is that a is five because that’s the constant real number.
That’s the coefficient on this x to the fourth term. So we’ve got a and f of x is this x to the fourth here.
So if we want to take the derivative of five x to the fourth, this formula for the derivative of a linear combination tells us that
we can bring the five out in front and just find the derivative of x to the fourth.
We just take the derivative of x to the fourth. Now, it’s a matter of just using power rule to find the derivative of x to the fourth.
So remember that to find the derivative of x to the fourth, we just multiply this coefficient out in front.
So we bring the four down in front, and then we leave the variable there x and subtract one from the exponent.
So four minus one is three. And if we simplify this by multiplying by the constant a which we factored out, five times four is twenty.
So we end up with twenty x cubed. So that’s all it’s saying.
Now let’s just take a look at a couple more examples to make sure we understand.
Again, negative six will be a; and f of x is just going to be this x term here.
So we can factor out the negative six and just take the derivative of x. Well, the derivative of x is just one.
So we end up with negative six times one, which of course just simplifies to negative six.
If we look at ninety times x raised to the one hundred, we can factor out the ninety
because this part right here is a according to our formula and this x to the one hundred here is f of x.
So if we factor out the ninety and we multiply by the derivative of x raised to the power of one hundred.
We leave this ninety out in front.
Remember to find the derivative of x to the one hundred,
we’ll bring the exponent out in front so we get one hundred out in front,
leave the variable x and then subtract one from the exponent so one hundred minus one is ninety-nine
So that’s the derivative of x to the one-hundred,
and then we just multiply it together, here, this ninety times one hundred, which gives us nine thousand.
So we end up with a final answer of nine thousand x to the ninety-nine.
So that’s it. That’s all this formula tells us in terms of how to find
the derivative of linear combination and you can do this with polynomials.
So for example, if you had these added together, if your problem was ninety x to the one hundred plus the first example we did,
five x to the four, you could do this one at a time.
Take the derivative of this first term and then the derivative of the second term and just add them together.
So you would end up with, if we bring our answer down here,
you would end up with nine thousand x to the ninety-nine plus twenty x cubed.
And that would be your final answer for this problem. So you can use it with polynomials.
You just apply it to one term at a time.
So that’s it. I hope this video helped you guys and I will see you in the next one.
Bye!