How would you differentiate each function?
1.) v(t)= sin^2 (sqrt (t))
2.) h(x)= sin x sin 2x sin 3x
2) h= uvw
h'= uvw'+uwv'+wvu'
1) v'= 2sin ( )cos ( ) 1/2sqrtt
To differentiate each function, you can use the chain rule and product rule respectively. Let's go through each function and explain how to differentiate them step by step.
1.) v(t) = sin^2(sqrt(t))
To differentiate this function, you need to apply the chain rule. The chain rule states that if you have a composition of functions, you need to differentiate the outer function and multiply it by the derivative of the inner function.
Step 1: Start by differentiating the outer function, sin^2(x). The derivative of sin^2(x) is 2sin(x)cos(x).
Step 2: Now differentiate the inner function, sqrt(t). The derivative of sqrt(t) is 1/(2*sqrt(t)).
Step 3: Apply the chain rule by multiplying the derivative of the outer function with the derivative of the inner function.
v'(t) = 2sin(sqrt(t))cos(sqrt(t)) * 1/(2*sqrt(t))
= sin(sqrt(t))cos(sqrt(t))/sqrt(t)
2.) h(x) = sin(x)sin(2x)sin(3x)
To differentiate this function, you need to apply the product rule. The product rule states that if you have a multiplication of functions, you need to differentiate each term and combine them using the product rule formula.
Step 1: Differentiate the first term, sin(x). The derivative of sin(x) is cos(x).
Step 2: Keep the first term unchanged and differentiate the second term, sin(2x). The derivative of sin(2x) is 2cos(2x).
Step 3: Keep the first two terms unchanged and differentiate the third term, sin(3x). The derivative of sin(3x) is 3cos(3x).
Step 4: Apply the product rule formula, which states (u'v + uv') for two functions u and v.
h'(x) = sin(x) * 2cos(2x) * sin(3x) + sin(x) * sin(2x) * 3cos(3x)
= 2sin(x)cos(2x)sin(3x) + 3sin(x)sin(2x)cos(3x)
And that's how you differentiate each function using the chain rule and product rule.