find the 8th derivative ofe^2x*sin2x?
The first derivative is
2 e^2x*sin2x + 2e^2x*cos2x
The second derivative is
2(2 e^2x*sin2x + 2e^2x*cos2x)
+ 2(2e^2x*cos2x -2e^2xsin2x)
= 8 e^2x*cos2x
Look for a pattern. The fourth derivative might be 8^4 times the original function. Then that would repeat after the eighth derivative
help me with this find higher derivative (d^2 y)/(dx^2 ) y=2x^5+3sin2x-4x+cos2x+ln2x^5
To find the 8th derivative of the function e^2x * sin(2x), we will need to use the product rule and the chain rule multiple times. Here's the step-by-step process:
Step 1: Start by writing out the original function: f(x) = e^2x * sin(2x).
Step 2: Apply the product rule to differentiate the function. The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by (u'v + uv').
Let's define u(x) = e^2x and v(x) = sin(2x).
The derivative of u(x) with respect to x, denoted as u'(x), is 2e^2x.
The derivative of v(x) with respect to x, denoted as v'(x), is 2cos(2x) (by applying the chain rule).
Applying the product rule, we get:
f'(x) = u'(x)v(x) + u(x)v'(x)
= 2e^2x * sin(2x) + e^2x * 2cos(2x).
Step 3: Repeat this differentiation process multiple times until we reach the 8th derivative.
Performing the second derivative, we use the product rule again:
f''(x) = [2e^2x * sin(2x) + e^2x * 2cos(2x)]' using the same steps as before.
Taking the derivative of 2e^2x * sin(2x), we get:
[2e^2x * sin(2x)]' = 4e^2x * sin(2x) + 2e^2x * 2cos(2x).
The derivative of e^2x * 2cos(2x) is found using the product and chain rule:
[e^2x * 2cos(2x)]' = 2e^2x * 2cos(2x) - e^2x * 4sin(2x).
Combining the two results, we have:
f''(x) = (4e^2x * sin(2x) + 2e^2x * 2cos(2x)) + (2e^2x * 2cos(2x) - e^2x * 4sin(2x))
= 6e^2x * sin(2x).
We need to repeat these steps 6 more times to find the 8th derivative. Perform the process for each derivative, simplifying and applying the chain rule as necessary.
f'''(x) = 12e^2x * cos(2x)
f''''(x) = -24e^2x * sin(2x)
f'''''(x) = -48e^2x * cos(2x)
f''''''(x) = 96e^2x * sin(2x)
f'''''''(x) = 192e^2x * cos(2x)
f''''''''(x) = -384e^2x * sin(2x)
f'''''''''(x) = -768e^2x * cos(2x)
Therefore, the 8th derivative of e^2x * sin(2x) is -768e^2x * cos(2x).