Just got through with derivatives, power rule, and chain rule.
I am supposed to find the first and second derivatives of the function.
y = 4x/√x+1 and y = cos^2(x)
y' = [(x+1)^(1/2)(4)]-[(4x)(1/2(x+1)^(-1/2))(1)]
y' = [4(x+1)^(1/2)-4x]/[2(x+1)^(1/2)(x+1)]
y' = (2-4x)/(x+1)
y" = [(x+1)(-4)]-[(2-4x)(1)]/(x+1)^2
y" = -6/(x+1)^2
y' = (cosx)^2
y' = 2(cosx)(-sinx)
y" = (-2cosx)(cosx)+(-sinx)(0)(-sinx)
y" = -2cos^2(x)
I feel like I am struggling with the chain rule and am just practicing
y = 4 x (x+1)^-.5
y' = 4x[-.5(x+1)^-1.5] +(x+1)^-.5[4}
= -2x /[(x+1)(x+1)^.5] + 4/(x+1)^.5
= -2x /[(x+1)(x+1)^.5] + 4(x+1)/[(x+1)](x+1)^.5
= (2x+4)(x+1)^-1.5
now do second
y = cos^2 x
y' = 2 cos x (-sin x) agree
but then
y" = 2 [ cos x (-cos x) -sin x(-sin x) ]
=2 [ - cos^2 x + sin^2 x]
y" = (2x+4)(-1.5(x+1)^-2.5) + (x+1)^-1.5(2)
y" = (x+1)^-1.5[2+(2x+4)(-3/2(x+1))]
y" = (x+1)^-1.5[2+(-6(x+1)/2(x+1))]
y" = (x+1)^-1.5[2+(-3)]
y" = -(x+1)^-1.5
Also, thank you for your time!!
It seems like you're on the right track with finding the derivatives using the power rule and chain rule. Here's a breakdown of how to find the first and second derivatives of the given functions:
1. For the function y = 4x/√(x+1):
- To find the first derivative, apply the quotient rule, which is a combination of the power rule and chain rule.
- Start with the numerator, which is 4x. Apply the power rule: differentiate x with respect to x to get 1, and keep the constant 4.
- Now, focus on the denominator, which is the square root of (x+1). Apply the chain rule: differentiate the function inside the square root, which is (x+1), to get 1, and multiply it by the derivative of (x+1), which is (1/2)(x+1)^(-1/2) using the power rule.
- Multiply the numerator of the first part by the denominator of the second part, and subtract the numerator of the second part multiplied by the denominator of the first part.
- Simplify the expression by combining like terms and factor out the common term (x+1)^(1/2)(x+1).
- To find the second derivative, apply the quotient rule again.
- Start with the first derivative, which is (2-4x)/(x+1). Apply the quotient rule again to find the second derivative.
- Differentiate the numerator of the first part, which is -4, and differentiate the denominator of the first part, which is (x+1), to get (x+1)^2.
- Differentiate the numerator of the second part, which is (2-4x), to get -4, and differentiate the denominator of the second part, which is (x+1), to get 1.
- Multiply the numerator of the first part by the denominator of the second part and subtract the numerator of the second part multiplied by the denominator of the first part.
- Simplify the expression and you'll get -6/(x+1)^2.
2. For the function y = cos^2(x):
- To find the first derivative, apply the chain rule.
- Start with the function (cosx)^2. Apply the power rule: differentiate cosx with respect to x, which gives -sinx, and keep the power of 2.
- Multiply the derivative of cosx by the power of 2, and simplify the expression to get -2cosxsinx or -2cos(x)sin(x).
- To find the second derivative, apply the chain rule again.
- Start with the first derivative, which is -2cos(x)sin(x). Apply the chain rule again to find the second derivative.
- Differentiate the first term, which is -2cos(x), to get 2sin(x), and differentiate the second term, which is sin(x), to get cos(x).
- Multiply the two terms and simplify the expression to get -2cos^2(x).
Remember, practice is key, and it's great that you're actively working on improving your understanding of the chain rule. Keep practicing and asking questions to deepen your understanding.