Find the derivative of the function.
g(u) = (5+u^2)^5(3-9u^2)^8
Could someone please explain the steps that would lead me to the answer? I'm completely stuck.
We can solve this by differentiation of a function of a function.
Given
g(u) = (5+u^2)^5(3-9u^2)^8
let
p(u)=(5+u²), and
q(u)=(3-9u²)
Then
g(u)=p(u)^5 * q(u)^8
Using the chain rule, we get
d(p(u)^5)/du
=5p(u)^4*dp(u)/du
=5p(u)^4*2u ...(1)
Similarly,
d(q(u)^8)/du
=8q(u)^7*d(q(u))/du
=9q(u)^7*(-18u) ... (2)
Now apply the product rule to the original function g(u):
d(g(u))/du
=(p(u)^5*d(q(u))/du + d(p(u))/du * q(u)^8
Substitute (1) and (2) and simplify to get:
13122*u*(u^2+5)^4*(3*u^2-1)^7*(39*u^2+115)
or some equivalent expression.
Go to : calc101 com
Click option derivatives
In rectacangle:Take the derivative of
type:
((5+u^2)^5)((3-9u^2)^8)
In rectacangle:with respect to:
type u
In rectacangle:and again with respect to
type u
Then click option: DO IT
You will see solution step-by-step
By the way on this site you can practice any kind of derivation.
To find the derivative of the function g(u) = (5+u^2)^5(3-9u^2)^8, we can use the product rule, which states that the derivative of a product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.
Here are the steps to find the derivative:
1. Start by identifying the two functions that are being multiplied together: (5+u^2)^5 and (3-9u^2)^8.
2. Next, we'll find the derivative of each individual function using the chain rule. The chain rule states that the derivative of a composite function is equal to the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function.
To find the derivative of (5+u^2)^5, we can let v = 5 + u^2. Now, we have v^5. To find the derivative of this function, we need to use the chain rule. The derivative of v^5 is 5v^4 multiplied by the derivative of v with respect to u. But v = 5 + u^2, so we need to find the derivative of v = 5 + u^2 with respect to u. Using the power rule, the derivative of u^2 is 2u. Thus, the derivative of v with respect to u is 2u.
Putting it all together, the derivative of (5+u^2)^5 is (5+u^2)^4 * 2u.
Now, let's find the derivative of the second function, (3-9u^2)^8. Similar to the previous function, we'll let w = 3 - 9u^2. Now, we have w^8. Again, we'll apply the chain rule. The derivative of w^8 is 8w^7 multiplied by the derivative of w with respect to u. The derivative of w = 3 - 9u^2 with respect to u is -18u. Therefore, the derivative of w with respect to u is -18u.
Combining this, the derivative of (3-9u^2)^8 is (3-9u^2)^7 * (-18u).
3. Now, we can apply the product rule. The product rule states that the derivative of a product of two functions is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.
So, applying the product rule, the derivative of g(u) is:
g'(u) = (5+u^2)^4 * 2u * (3-9u^2)^8 + (5+u^2)^5 * (3-9u^2)^7 * (-18u).
Simplifying this expression will give you the final result, which is the derivative of the function g(u).