Find the derivative.
StartFraction d Over dt EndFraction Integral from 0 to t Superscript 8 StartRoot u cubed EndRoot du
a. by evaluating the integral and differentiating the result.
b. by differentiating the integral directly.

Using either method, we obtain:  

Step-by-step explanation:

a) By evaluating the integral:

The integral itself can be evaluated by writing the root and exponent of the variable u as:  

Then, an antiderivative of this is:

which evaluated between the limits of integration gives:

and now the derivative of this expression with respect to "t" is:

b) by differentiating the integral directly: We use Part 1 of the Fundamental Theorem of Calculus which states:

"If f is continuous on [a,b] then

is continuous on [a,b], differentiable on (a,b) and  

Since this this function is continuous starting at zero, and differentiable on values larger than zero, then we can apply the theorem. That means:

2nd one in the first row is correct

y = 2x

step-by-step explanation:

if you divide each number in the sequence, you'll see that they all have a multiple factor of 2. each number in the series, and any other numbers which may continue, will be multiplied by 2. therefore, y is the next number in the sequence, and x is the current one, so, our formula will be y = 2x.