What would an example of a logarithmic function that contains two trig functions within it where you would need to use chain rule and product rule to take derivative look like? How would taking the derivative of the function in its original form look different when using log properties to simplify the function first and then take derivative?
An example of a logarithmic function that contains two trigonometric functions within it, and where you would need to use the chain rule and product rule to take the derivative, could be:
f(x) = ln(sin(x)cos(x))
To apply the chain rule, we need to identify the inner and outer functions. In this case, the outer function is the natural logarithm (ln), and the inner function is the product of sine (sin(x)) and cosine (cos(x)).
To calculate the derivative of this function using the chain rule, we start by differentiating the outer function and then multiplying it by the derivative of the inner function:
f'(x) = (1/sin(x)cos(x)) * (cos(x)cos(x) - sin(x)sin(x))
f'(x) = (cos^2(x) - sin^2(x)) / (sin(x)cos(x))
Now, let's consider simplifying the function by using log properties before taking the derivative. We know that ln(a * b) is equivalent to ln(a) + ln(b), so we can rewrite the original function as:
f(x) = ln(sin(x)) + ln(cos(x))
Taking the derivative of this simplified form is more straightforward. We can differentiate each term separately:
f'(x) = (1/sin(x)) * cos(x) + (1/cos(x)) * (-sin(x))
f'(x) = cos(x)/sin(x) - sin(x)/cos(x)
As you can see, the simplified form of the derivative is mathematically equivalent to the earlier calculated derivative using the chain rule. However, using log properties to simplify the function first can sometimes make the calculation easier and more efficient.