Learn How To Use The Chain Rule For Finding Derivatives With This Step-By-Step Guide

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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