Find the derivatives of these:
f(x)=(√(x))sinPix
Also:
f(x)=√(Cosx+sinx)
Steps would be greatly helpful as I will be able to use them as reference. Thank you.
f'(x)=(√(x))'*(sin(pix))+(sin(pix))'*(√(x))=1/2*(x)^(-1/2)*(sin(pix))+cos(pix)*(pi)*(√(x))
2)
f'(x)=(1/2(cosx+sinx)^(-1/2))(cos(x)+sin(x))'=(1/2(cosx+sinx)^(-1/2)*(-sin(x)+cos(x))
correct answers, but they're sort of like leaving an answer as 4x^2y/3xy instead of simplifying to 4x/3
1)1/2√x sinπx + π/√x cosπx
= 1/2√x (sinπx + 2πcosπx)
2)
(cosx-sinx) / 2√(cosx+sinx)
To find the derivatives of the given functions, we can use the rules of differentiation. Here's a step-by-step explanation for each function:
1. f(x) = (√(x))sin(Pix)
Step 1: Identify the composite function. In this case, we have the product of two functions: (√(x)) and sin(Pix).
Step 2: Differentiate each individual function.
- To differentiate (√(x)), we can rewrite it as x^(1/2), then apply the power rule. The derivative will be (1/2)x^(-1/2).
- To differentiate sin(Pix), we need to apply the chain rule. The derivative of sin(u) is cos(u), and the derivative of u with respect to x is du/dx. In this case, u = Pix, so the derivative of sin(Pix) becomes cos(Pix) * d(Pix)/dx.
Step 3: Multiply the derivatives obtained from step 2 by their respective functions.
f'(x) = [(1/2)x^(-1/2)] * sin(Pix) + (√x) * [cos(Pix) * d(Pix)/dx]
Simplifying this expression further may not be possible without more specific values. However, these are the necessary steps for finding the derivative of f(x) = (√(x))sin(Pix).
2. f(x) = √(cos(x) + sin(x))
Step 1: Identify the composite function. In this case, we have √(cos(x) + sin(x)).
Step 2: Differentiate the composite function using the chain rule.
- Recall that the chain rule states that if we have a composite function u(v(x)), the derivative is given by du/dx = du/dv * dv/dx.
For our function f(x) = √(cos(x) + sin(x)), let's set u = √v, where v = cos(x) + sin(x).
- Derivative of u = 1/(2√v), as per the power rule for square roots.
- Derivative of v = derivative of (cos(x) + sin(x)), which is -sin(x) + cos(x).
Step 3: Multiply du/dv by dv/dx.
f'(x) = (1/(2√v)) * (-sin(x) + cos(x))
Now, substitute back v = cos(x) + sin(x).
f'(x) = (1/(2√(cos(x) + sin(x)))) * (-sin(x) + cos(x))
These steps outline the process of finding the derivative of f(x) = √(cos(x) + sin(x)).