Patrick C. answered • 08/19/21
Hi Ana,
In order to answer this question, we will need to use the Chain Rule for derivatives since we have a composite function. The way that I like to break down these problems is by identifying the inner and outer functions first and labeling them as g(x) and f(x) respectively.
In this case, if f(g(x)) = tan(3x+7), then
g(x) = 3x+7
f(x) = tan(x)
Next, we need to find the derivative of the inner function, g'(x), the derivative of the outer function, f'(x) and plug in the original inner function into the derivative of the outer function to find f'(g(x)).
Using the Power Rule and knowing that the derivative of a constant is 0,
then g'(x) = 3
Using our trigonometric derivatives, the derivative of tangent is secant squared so
f'(x) = sec^2(x)
Next, we plug in the original g(x), (3x+7), into the f'(x) equation to get that f'(g(x)) = sec^2(3x+7).
From the Chain Rule, if we take the derivative of f(g(x)), then we end up with f'(g(x)) * g'(x).
Thus,
d/dx((tan(3x+7)) = sec^2(3x+7) * 3
Therefore, if we rearrange the left hand side of our equation, then dy/dx = 3 * sec^2(3x+7).
Lastly, to find the differential dy, then we multiply both sides by dx to get
dy = 3 sec^2(3x+7) dx.
You can then plug in your corresponding values of x and dx into that equation to find your final answer.
