How do you use the chain rule to differentiate #g(x)=3tan(4x)#?

Question

Isaiah Allen · Accepted Answer

#g'(x)=12sec^2(4x)# To differentiate using the #color(blue)"chain rule"# #color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(g(h(x)))=g'(h(x))h'(x))color(white)(a/a)|)))......(A)# #color(blue)"----------------------------------------------------"# #g(h(x))=3tan(4x)rArrg'(h(x))=3sec^2(4x)# and #h(x)=4xrArrh'(x)=4# #color(blue)"-----------------------------------------------------"# substitute these values into (A) #rArrg'(x)=3sec^2(4x).4=12sec^2(4x)#

Evelyn Agard · Answer

undefined To differentiate $ g(x) = 3 \tan(4x) $ using the chain rule, follow these steps:

1. Identify the outer function, which is $ \tan(4x) $, and the inner function, which is $ 4x $.
2. Find the derivative of the outer function with respect to its inner variable, which is $ \frac{d}{dx}[\tan(u)] = \sec^2(u) \frac{du}{dx} $, where $ u = 4x $.
3. Calculate the derivative of the inner function with respect to $ x $, which is simply $ 4 $.
4. Substitute the results from steps 2 and 3 into the chain rule formula.
5. Multiply the derivative of the outer function by the derivative of the inner function to obtain the derivative of $ g(x) $.

Following these steps, the derivative of $ g(x) $ with respect to $ x $ is:

$$ g'(x) = 3 \sec^2(4x) \cdot 4 $$