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
It seems the post didn't post so I put it again.
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
Sure! I can help you with finding the derivatives of these functions. Here are the detailed steps for each problem:
1) To find the derivative of y = cot(x) + sin(x), we'll use the sum rule for derivatives. First, we find the derivative of cot(x) and sin(x) separately. The derivative of cot(x) is -csc^2(x), and the derivative of sin(x) is cos(x). Then, we simply add these derivatives together to get the derivative of y.
2) To find the derivative of y = (3x + 5)^8, we'll use the chain rule. First, we calculate the derivative of the inner function, which is 3x + 5. The derivative of 3x + 5 with respect to x is simply 3. Then, we multiply this derivative by the derivative of the outer function, which is 8(3x + 5)^7. So the derivative of y is 8(3x + 5)^7 * 3.
3) To find the derivative of y = x csc(x), we'll also use the product rule. The derivative of x is 1, and the derivative of csc(x) is -csc(x)cot(x). Then, we multiply these derivatives together using the product rule formula, giving us the derivative of y.
4) To find the derivative of y = √(x √(3x+1)), we'll use the chain rule. First, we find the derivative of the innermost function, 3x + 1. The derivative of 3x + 1 with respect to x is 3. Then, we multiply this derivative by the derivative of the next function, √(3x + 1), which is (1/2)√(3x + 1). Finally, we multiply this derivative by the derivative of the outer function, √(x √(3x + 1)), which is (1/2) * √(3x + 1). So the derivative of y is (1/2) * √(3x + 1) * (1/2)√(3x + 1) * 3.
5) To find (f ◦ g)(x) if f(x) = x^5 + 1 and g(x) = √x at x = 1, we first plug g(x) into f(x), which gives us f(g(x)) = (√x)^5 + 1. Simplifying this, we have f(g(x)) = x^(5/2) + 1. Finally, we evaluate f(g(x)) at x = 1 to get the final answer.
I hope these step-by-step explanations help you understand how to find the derivatives and solve these problems. Let me know if you have any further questions!