use f'(x)=lim h-->0 f(x+h)-f(x)/h to find the limit: h-->0 sin^3,4(x+h)-sin^3,4x/h
To find the limit of the expression h --> 0 [(sin^3(4(x+h)) - sin^3(4x)) / h] using the definition of the derivative, we can apply the limit definition of the derivative and simplify the expression.
Starting with the given expression:
f'(x) = lim (h --> 0) [sin^3(4(x+h)) - sin^3(4x)] / h
Let's expand the cube terms using the identity (a^3 - b^3) = (a - b)(a^2 + ab + b^2):
f'(x) = lim (h --> 0) [(sin^3(4x + 4h) - sin^3(4x))] / h
= lim (h --> 0) [(sin(4x + 4h) - sin(4x))(sin^2(16x + 16h) + sin(16x + 16h)sin(16x) + sin^2(16x))] / h
Now, let's focus on the first term:
lim (h --> 0) [sin(4x + 4h) - sin(4x)] / h
Using the limit definition of the derivative, we have:
f'(x) = lim (h --> 0) [sin(4x + 4h) - sin(4x)] / (4h)
Next, we can simplify the expression by applying the identity sin(a+b) - sin(a) = 2sin((a+b)/2)cos((a-b)/2):
f'(x) = lim (h --> 0) [2sin((4x + 4h + 4x) / 2)cos((4x + 4h - 4x) / 2)] / (4h)
= lim (h --> 0) [2sin((8x + 4h) / 2) cos(2h)] / (4h)
Simplifying further:
f'(x) = lim (h --> 0) [2sin(4x + 2h)cos(2h)] / (4h)
= lim (h --> 0) [2sin(4x + 2h)] / (2h) * (cos(2h) / 2)
Now, we can evaluate the limit as h approaches 0:
f'(x) = lim (h --> 0) sin(4x + 2h) / h * cos(2h) / 2
= 2cos(0) * cos(0) / 2
= 1
Therefore, the limit of the expression is 1.