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 �ã3x + 1 (3x + 1 is under the square root)
5) ( f ◦ g )�Œ if f ( x) = x^5 + 1 and g (x) = �ãx at x = 1 (the x is under the square root)
Best Regards
2) Let u = 3x+5 and use the chain rule
dy/dx = dy/du*du/dx
y = u^8
dy/du = 8 u^7
u = 3x+5
du/dx = 3
You complete the problem. If you need help with the others, show your work and help will be provided.
Sure, I would be happy to help you with each of these problems. Here are the steps to find the derivatives for each problem:
1) To find the derivative of y = cot x + sin x, we can use basic derivative rules. The derivative of cot x is -csc^2 x, and the derivative of sin x is cos x. So, the derivative of y with respect to x is -csc^2 x + cos x.
2) To find the derivative of y = (3x+5)^8, we can use the chain rule. The derivative of (3x+5) with respect to x is 3. Then, we multiply it by the derivative of (3x+5)^8 with respect to (3x+5), which is 8(3x+5)^(8-1). Therefore, the derivative of y with respect to x is 24(3x+5)^7.
3) To find the derivative of y = x csc x, we can use both the product rule and the chain rule. The derivative of x with respect to x is 1. The derivative of csc x with respect to x is -csc x cot x. Using the product rule, we have 1 * csc x + x * (-csc x cot x). Simplifying further, we can rewrite it as csc x - x csc x cot x.
4) To find the derivative of y = sqrt(x^(3x + 1)), we can use the chain rule. The derivative of sqrt(x) is 1/(2 sqrt(x)). Then, we multiply it by the derivative of x^(3x + 1) with respect to x, which is (3x + 1) * x^(3x) + x^(3x + 1) * ln(x). Therefore, the derivative of y with respect to x is (3x + 1) * x^(3x) + x^(3x + 1) * ln(x) * (1/(2 sqrt(x))).
5) To find (f ◦ g)(x) if f(x) = x^5 + 1 and g(x) = sqrt(x) at x = 1, we need to substitute g(x) into f(x) and evaluate it at x = 1. So, (f ◦ g)(x) becomes f(g(x)) = f(sqrt(x)). Substituting g(x) into f(x), we get (sqrt(x))^5 + 1 = x^(5/2) + 1. Evaluating at x = 1, we have (f ◦ g)(1) = 1^(5/2) + 1 = 1 + 1 = 2.
I hope this helps! If you have any further questions, feel free to ask.