Let f be a differentiable function. If h(x)=(1+f(3x))2, which of the following gives a correct process for finding h′(x) ?
Correct. This requires using the chain rule twice to compute the derivative. h′(x)=2(1+f(3x))⋅(ddx(1+f(3x)))=2(1+f(3x))(f′(3x)⋅ddx(3x))=2(1+f(3x))⋅f′(3x)⋅3
We can use the chain rule to find h′(x):
h(x) = (1 + f(3x))²
Let u = 1 + f(3x)
Then h(x) = u²
Applying the chain rule, we have:
h′(x) = 2u * u′
Now we need to find u′:
u = 1 + f(3x)
So, u′ = f′(3x) * 3
Putting it all together, we have:
h′(x) = 2(1 + f(3x)) * f′(3x) * 3
Simplifying, we get:
h′(x) = 6(1 + f(3x)) * f′(3x)
Therefore, the correct process for finding h′(x) is:
h′(x) = 6(1 + f(3x)) * f′(3x)
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