okay i need to find the derivative of
f(x)=πsin2x
and
f(x)=cosπx^2
----------------
π is pi
and if you could show steps it would be GREATLY appreciated. thank you so much!
you need to use the product rule so the first one(which is pi) times the derivative of the second one(which is sin2x) plus the second one times the derivative of the first one.
Here is how it should look:
(pi*cos2x)+(sin2x*1)The one is because pi is a constant so its derivative is just one.
In the second problem I do not understand what is squared...the x or pi times x. Sorry
I hope I helped you one the first one!
Hey, thank you so much! you did help!
the second is x squared.. sorry for no being clear
π is just a constant, so
d/dx(πx) = π
d/dx(πsin2x) = π(2cos2x) = 2πcos2x
the second one uses the chain rule:
if u is a function of x, and f = cos(u), f' = -sin(u) u'
u = πx^2, so u' = 2πx
d/dx cos(πx^2) = -2πx sin(πx^2)
PS - derivative of a constant is 0, not 1.
Reason 1: derivative is rate of change. constants do not change.
Reason 2: since 1 = x^0, a constant k = kx^0
the derivative is thus 0*k*x^(-1) = 0
Sure! I'd be happy to help you find the derivatives of the given functions using the rules of differentiation.
1. f(x) = πsin^2x
To find the derivative of this function, we can use the chain rule. The chain rule states that if we have a function in the form f(g(x)), the derivative is given by f'(g(x)) * g'(x).
Let's apply the chain rule step by step:
Step 1: Identify f(u) and g(x).
- f(u) = u^2, where u = sin(x)
- g(x) = πx
Step 2: Find the derivatives of f(u) and g(x).
- f'(u) = 2u
- g'(x) = π
Step 3: Apply the chain rule.
- f'(g(x)) = f'(u) * g'(x)
- f'(g(x)) = 2u * π
Step 4: Substitute u = sin(x).
- f'(g(x)) = 2sin(x) * π
Therefore, the derivative of f(x) = πsin^2x is f'(x) = 2πsin(x).
2. f(x) = cos(πx^2)
To find the derivative of this function, we can again use the chain rule.
Step 1: Identify f(u) and g(x).
- f(u) = cos(u), where u = πx^2
- g(x) = πx^2
Step 2: Find the derivatives of f(u) and g(x).
- f'(u) = -sin(u)
- g'(x) = 2πx
Step 3: Apply the chain rule.
- f'(g(x)) = f'(u) * g'(x)
- f'(g(x)) = -sin(u) * 2πx
Step 4: Substitute u = πx^2.
- f'(g(x)) = -sin(πx^2) * 2πx
Therefore, the derivative of f(x) = cos(πx^2) is f'(x) = -2πx * sin(πx^2).
I hope this explanation helps! Let me know if you have any further questions.