Differentiate √(30+sin²x) with respect to x
let p = (30+sin²x)
then we want
d/dx ( p^.5) = .5 p^-.5 dp/dx
but dp/dx = 2 sin x cos x
so
.5 (1/p^.5) 2 sin x cos x
= sin x cos x / √(30+sin²x)
Apologies for the mistake in my previous response. You are correct.
Let p = (30+sin²x).
We want to find d/dx (p^.5).
Using the chain rule, we have:
d/dx (p^.5) = .5 p^(-.5) dp/dx
Now, let's find dp/dx.
dp/dx = d(30+sin²x)/dx = 2sinx*cosx
Substituting this back into the previous equation, we have:
.5 p^(-.5) dp/dx = .5 (1/√p) * 2sinx*cosx
Simplifying further:
= sinx*cosx / √(30+sin²x)
Therefore, the correct differentiated form of √(30+sin²x) with respect to x is sinx*cosx / √(30+sin²x).
To differentiate √(30+sin²x) with respect to x, we can use the chain rule. Let's break down the steps:
Step 1: Identify the composite function:
In this case, the composite function is √(30+sin²x).
Step 2: Differentiate the outer function:
The outer function is √( ) or the square root function. The derivative of the square root function is 1/(2√(x)). Applying this derivative to the outer function, the derivative becomes 1/(2√(30+sin²x)).
Step 3: Differentiate the inner function:
The inner function is 30+sin²x. We need to differentiate this with respect to x. The derivative of 30 with respect to x is 0, and the derivative of sin²x with respect to x is 2sin(x)cos(x) or 2sinxcosx.
Step 4: Apply the chain rule:
Now we multiply the derivative of the outer function by the derivative of the inner function.
The final answer is (1/(2√(30+sin²x))) * (2sinxcosx) = sinxcosx / √(30+sin²x).
Therefore, the differentiation of √(30+sin²x) with respect to x is sinxcosx / √(30+sin²x).
To differentiate √(30+sin²x) with respect to x, we can use the chain rule.
Let's denote the function as f(x) = √(30+sin²x).
To differentiate f(x) with respect to x, we differentiate the outer function (square root) and then multiply by the derivative of the inner function (30+sin²x).
The derivative of the square root function is 1/2√(30+sin²x).
The derivative of the inner function, 30+sin²x, with respect to x is 2cosx * (-sinx) = -2sinxcosx.
Multiplying these two results together, we have:
f'(x) = -2sinxcosx * 1/2√(30+sin²x)
Simplifying further:
f'(x) = -sinx * cosx * 1/√(30+sin²x)
Therefore, the differentiated form of √(30+sin²x) with respect to x is -sinx * cosx / √(30+sin²x).