how to find the derivative of the function. y=integral cos(u^4) from 4cos(x) to 4x
integral cos u^4) du = (1/4) sin u^4
at u = 4x integral = (1/4) sin (4x)^4
at u = 4 cos x integral = (1/4) sin (4 cos x)^4
so definite integral = (1/4)[ sin (4x)^4 - sin (4 cos x)^4 ]
d/dx of that
= (1/4)[ 16 sin (4x)^3 - 4 sin(4cosx)^3 (-4sin x) ]
etc
The derivative F'(x) of the integral
F(x) = ∫[a(x),b(x)] f(u) du = F(b(x))*b'(x) - f(a(x))*a'(x)
with
a(x)=cos(4x)
b(x)=4x
f(u)=cos(u^4)
F'(x)= cos((4cosx)^4)*(4cosx)^3*(-4sinx) - (cos(4x)^4)*4(4x)^3*4
To find the derivative of the function, we will use the Fundamental Theorem of Calculus.
Let's denote the integral as a function of x, denoted by F(x):
F(x) = ∫[4cos(x), 4x] cos(u^4) du
To find the derivative of F(x), we will consider the upper limit of integration as a function of x (4x), and then apply the chain rule.
First, we apply the chain rule to find the derivative of the upper limit of integration (4x).
d/dx (4x) = 4
Next, we substitute the derivative back into the integral:
d/dx F(x) = d/dx ∫[4cos(x), 4x] cos(u^4) du
= d/du ∫[4cos(x), 4x] cos(u^4) du * d/dx (4x)
= cos((4x)^4) * 4
Simplifying further:
d/dx F(x) = 4cos((4x)^4)
Therefore, the derivative of the given function with respect to x is 4cos((4x)^4).
To find the derivative of the function, we need to apply the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus states that if "F(x)" is the integral of a function "f(x)" from "a" to "x", then the derivative of "F(x)" with respect to "x" is equal to "f(x)".
In this case, we have y = ∫cos(u^4) du, where the limits of integration are from 4cos(x) to 4x.
To find the derivative of y with respect to x, we will differentiate the integral function, using the chain rule.
Let's go step by step:
Step 1: Define a new function F(u) = ∫cos(u^4) du.
Step 2: Apply the Fundamental Theorem of Calculus. According to the theorem, the derivative of F(u) with respect to u is equal to the integrand, which is cos(u^4). Therefore, we have:
dF/du = cos(u^4).
Step 3: Apply the chain rule to find dy/dx. The chain rule states that if y = F(g(x)), then dy/dx = dF/dg * dg/dx. Applying this rule to our case,
dy/dx = dF/du * du/dx.
Step 4: Now, we need to find du/dx. Notice that in the original integral, the variable of integration is u, not x. So we need to express u in terms of x.
Given that u = 4x, by differentiating both sides with respect to x, we have du/dx = 4.
Step 5: Substitute the values we found into the chain rule expression:
dy/dx = dF/du * du/dx = cos(u^4) * 4.
Step 6: Substitute u = 4x back into the equation:
dy/dx = 4 * cos((4x)^4) = 4 * cos(256x^4).
So, the derivative of y with respect to x is given by 4 * cos(256x^4).