Differntiate
sin(sinx) using the product rule
y = sin(sin x)
y' = cos (sin x) * cos x
how can you explain the steps and formula
Let z = sin x
then we have
d/dx [ sin z }
which is
cos z * dz/dx
or
cos (sin x) * dz/dx
but dz/dx is cos x
so
cos (sin x) * cos x
To differentiate the function sin(sinx) using the product rule, we need to consider that sin(x) is the first function and sin(sinx) is the second function. Let's denote the first function as u(x) = sin(x) and the second function as v(x) = sin(sinx).
Now, let's calculate the derivatives of u(x) and v(x):
The derivative of u(x) = sin(x) with respect to x is du/dx = cos(x).
The derivative of v(x) = sin(sinx) with respect to x is a little more complicated. We need to apply the chain rule.
Let's break it down:
- Let y = sin(x)
- Now, we can rewrite the function v(x) = sin(sinx) as v(y) = sin(y).
Now, using the chain rule:
dv/dx = dv/dy * dy/dx.
dv/dy represents the derivative of v(y) with respect to y, and dy/dx represents the derivative of y with respect to x.
Calculating them:
- dv/dy = cos(y)
- dy/dx = cos(x) since y = sin(x)
Now, substituting these values back into the chain rule equation:
dv/dx = cos(y) * cos(x).
Finally, substituting y = sin(x) back into the equation:
dv/dx = cos(sin(x)) * cos(x).
So, the derivative of sin(sinx) using the product rule is du/dx * v(x) + u(x) * dv/dx, which becomes:
cos(x) * sin(sinx) + sin(x) * cos(sin(x)) * cos(x).
Therefore, the differentiation of sin(sinx) using the product rule is cos(x) * sin(sinx) + sin(x) * cos(sin(x)) * cos(x).