Any help would be much appreciated with the steps involved in each problem given. Thank you.
1) Find derivative if y =cot x + sin x
2) Find derivative if y = (3x+5)^8
3) Find derivative if y = x csc x
4) Find derivative if y = x /3x + 1 (3x + 1 is under a square root(/)
5) ( f ◦ g )�Œ if f (x) = x^5 + 1 and g (x) = /x at x = 1 (the x is under a square root(/)
Best Regards
2) use the chain rule where 2 separated parts.
y= (3x+ 5)^8
x= 3x+5
dx= 3
y'= x^8dx
8x^7 dx
put back what x is and get
8*3(3x+5)^7
I just put back the dx as well and moved it to multiply it with the 8
so..
24(3x+5)^7
this is what it should be if I remember how to do this since I'm not looking at my cal book right now.
1) there's an identity for d/dx (cot x) which = csc^2 x. So the answer to 1) is csc^2 x + cos x.
3) use the derivative multiplying rule first *d(second) + second *d(first):
y' = x*(-csc(x)*cot(x)) + csc(x)*1
-- there may be some simplification you can do, but I don't know how much that part matters.
4) remember that the square root of 'something' = (something)^(1/2), so you can rewrite the equation as y = x / (3x+1)^(1/2)
Use the dividing derivative rule of [bottom * d(top) - top * d(bottom)] / bottom^2.
[(3x+1)^(1/2)*1 - x(3x+1)^(1/2)] / {(3x+1)^(1/2)}^2
[(3x+1)^(1/2) - x*(3x+1)^(1/2)] / (3x+1)
5) not sure about the syntax on 5. It may be more clear if you use sqrt(x) or x^(1/2) to denote square roots.
Sure! I'd be happy to help you with each of these problems and explain the steps involved in finding the derivatives.
1) To find the derivative of y = cot(x) + sin(x), we can use the rules of differentiation. The derivative of cot(x) is -cosec^2(x) and the derivative of sin(x) is cos(x). Therefore, the derivative of y with respect to x (dy/dx) is -cosec^2(x) + cos(x).
2) To find the derivative of y = (3x+5)^8, we can use the chain rule. The chain rule states that if we have a function raised to a power, we multiply the derivative of the function by the exponent, and then multiply it by the derivative of the function itself. In this case, the derivative of (3x+5) is 3, and the derivative of 3x+5 is 3. Thus, the derivative of y with respect to x is 8(3x+5)^7 * 3.
3) To find the derivative of y = x csc(x), we can again use the product rule. The product rule states that if we have two functions multiplied together, the derivative of the product is the first function times the derivative of the second function, plus the second function times the derivative of the first function. In this case, the derivative of csc(x) is -csc(x) * cot(x), and the derivative of x is 1. Thus, the derivative of y with respect to x is xcsc(x) * (-csc(x) * cot(x)) + 1 * csc(x).
4) To find the derivative of y = x / √(3x + 1), we can use the quotient rule. The quotient rule states that if we have a fraction, the derivative is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator. In this case, the derivative of x is 1, and the derivative of √(3x + 1) is (3/2) * (1/√(3x + 1)). Thus, the derivative of y with respect to x is (√(3x + 1) - x * (3/2) * (1/√(3x + 1))) / (3x + 1).
5) For (f ◦ g)(x) = f(g(x)), we substitute the function g(x) into function f(x). In this case, f(x) = x^5 + 1 and g(x) = √x. Substituting g(x) into f(x) gives us f(g(x)) = (√x)^5 + 1 = x^(5/2) + 1. Therefore, (f ◦ g)(x) = x^(5/2) + 1. To find the value of (f ◦ g)(1), we substitute x = 1 into (f ◦ g)(x). So, (f ◦ g)(1) = (1)^(5/2) + 1 = 1 + 1 = 2.
I hope this helps! Let me know if you have any further questions.