Differentiate in terms of x:
1] y = cos^3 5x
2] y = sec^2 3x
start out this way
3 [cos^2(5x)][-sin(5x)](5)
I do not have derivative of sec in my head so:
2 [sec (3x)][d/dx(1/cos(3x)](3)
and
d/dx[1/cos(3x)] = [sin(3x)(3)]/cos^2(3x)
then combine
1] let u (x) = cos(5x)
y = u^3
dy/dx = dy/du*du/dx = 3 u^2*d(cos 5x)/dx
= 3 cos^2(5x)*[-sin(5x)]*5
= -15 cos^(5x)* sin(5x)
I used the "chain rule" a second time to get d(cos 5x)/dx
2] Proceed similarly. Use the chain rule twice.
Let u(x) = sec(3x)
f(x) = u^2
df/dx = df/du * du/dx
= 2 u * d/dx (sec(3x))
Let v = 3x dv/dx = 3
df/dx = 2 u (d/dv)(sec(v))*(dv/dx)
= 6 sec(3x)*d/dv(sec v)
= 6 sec(3x)*tan(3x)*sec*(3x)
= 6 sec^2(3x)*tan(3x)
Where I wrote f and df/dx, you can substitute y and dy/dx
To differentiate the given functions, we can use the chain rule. The chain rule states that if you have a composite function with an outer function and an inner function, the derivative is the derivative of the outer function multiplied by the derivative of the inner function.
Let's differentiate each of the given functions step by step:
1) To differentiate y = cos^3(5x), we can treat it as a composite function. The outer function is y = u^3, where u = cos(5x). The inner function is u = cos(5x).
Step 1: Find the derivative of the inner function.
du/dx = d/dx (cos(5x)) = -5 sin(5x)
Step 2: Find the derivative of the outer function.
dy/du = d/du (u^3) = 3u^2
Step 3: Apply the chain rule by multiplying the derivatives obtained from steps 1 and 2.
dy/dx = (dy/du)(du/dx) = 3(cos^2(5x))(-5sin(5x))
Therefore, the derivative of y = cos^3(5x) is dy/dx = -15(cos^2(5x))(sin(5x)).
2) To differentiate y = sec^2(3x), we apply a similar process:
Step 1: Find the derivative of the inner function.
du/dx = d/dx (sec(3x)) = 3 sec(3x) tan(3x)
Step 2: Find the derivative of the outer function.
dy/du = d/du (u^2) = 2u
Step 3: Apply the chain rule by multiplying the derivatives obtained from steps 1 and 2.
dy/dx = (dy/du)(du/dx) = 2(sec^2(3x))(3sec(3x)tan(3x))
Therefore, the derivative of y = sec^2(3x) is dy/dx = 6(sec^2(3x))(sec(3x)tan(3x)).
In summary:
1) y = cos^3(5x), dy/dx = -15(cos^2(5x))(sin(5x))
2) y = sec^2(3x), dy/dx = 6(sec^2(3x))(sec(3x)tan(3x))