Let y = sin(x^2-3). find dy.
dy = (dy/dx)*dx
= 2x*cos(x^2-3) dx
I used the "chain rule"
dy/dx = dy/du*du/dx, with
u = x^2 -3
y = sin u
To find dy, we need to differentiate the given equation with respect to x.
Let's break it down step-by-step:
Step 1: Identify the function to be differentiated.
In this case, the function to be differentiated is y = sin(x^2-3).
Step 2: Apply the chain rule.
The chain rule states that if we have a composition of functions, g(f(x)), then the derivative of this composition is given by g'(f(x)) * f'(x).
Step 3: Differentiate the outer function.
The outer function is sin(u), where u = x^2-3. The derivative of sin(u) with respect to u is cos(u).
Step 4: Differentiate the inner function.
The inner function is u = x^2-3. The derivative of u with respect to x is du/dx = 2x.
Step 5: Combine the results.
Using the chain rule, dy/dx = cos(u) * du/dx.
Substituting the values we found for cos(u) and du/dx, we have:
dy/dx = cos(x^2-3) * 2x.
Therefore, the derivative dy/dx of y = sin(x^2-3) is dy/dx = cos(x^2-3) * 2x.
To find the derivative of y = sin(x^2 - 3), we can use the chain rule. The chain rule states that if we have a composition of functions, like f(g(x)), where f(u) and g(x) are differentiable functions, then the derivative of f(g(x)) is given by f'(g(x))·g'(x).
First, let's find the derivative of the inner function, x^2 - 3, with respect to x. Since the derivative of x^2 is 2x, the derivative of x^2 - 3 with respect to x is 2x.
Next, we find the derivative of the outer function, sin(u), with respect to u. The derivative of sin(u) is cos(u).
Now, using the chain rule, we can find the derivative of y = sin(x^2 - 3) with respect to x:
dy/dx = cos(x^2 - 3) · (2x)
Therefore, the derivative of y = sin(x^2 - 3) is dy/dx = 2x · cos(x^2 - 3).