Find the first and second derivatives of the following function:
f(x)=7cosx+sin2x
well,
d/dx sinx = cosx
d/dx cosx = -sinx
so just crank 'em out
To find the first and second derivatives of the given function f(x) = 7cos(x) + sin(2x), we can follow these steps:
Step 1: Find the derivative of each term separately.
The derivative of cos(x) is -sin(x), and the derivative of sin(2x) can be found using the chain rule: the derivative of sin(u) is cos(u) * u' (where u is a function of x).
So, the derivative of sin(2x) will be cos(2x) * 2.
Step 2: Combine the derivatives.
The derivative of f(x) will be obtained by adding the derivatives of each term together:
f'(x) = -7sin(x) + 2cos(2x).
Step 3: Find the second derivative.
To find the second derivative, we need to differentiate f'(x).
The derivative of -7sin(x) is -7cos(x), and the derivative of 2cos(2x) can be found using the chain rule: the derivative of cos(u) is -sin(u) * u'.
So, the derivative of 2cos(2x) will be -2sin(2x) * 2.
Step 4: Simplify the second derivative.
Combining the derivatives, we get:
f''(x) = -7cos(x) - 4sin(2x).
Therefore, the first derivative of the given function is f'(x) = -7sin(x) + 2cos(2x), and the second derivative is f''(x) = -7cos(x) - 4sin(2x).