Question

Let g(x)= x⁵ - 3x and let h be the inverse function of g. notice that g(1)=4 h'(4)=

Answer 1

`\[ h'(4) = (1)/(g'(h(4))) \]`

The derivative of the inverse function
`\( h \) at \( x = 4 \) is equal to the reciprocal of the derivative of \( g \) at \( x = 1 \), which is \( (1)/(2) \).`

Step-by-step explanation:

The inverse function rule states that if
`\( h \)`is the inverse of
`\( g \)`, then the derivative of
`\( h \)`at a particular point is the reciprocal of the derivative of \( g \) at the corresponding point. In this case,
`\( g(1) = 4 \), so \( h(4) = 1 \)`. The derivative of
`\( g \) at \( x = 1 \) is \( g'(1) \). Therefore, the derivative of \( h \) at \( x = 4 \) is \( (1)/(g'(1)) \).`

Now, let's find
`\( g'(x) \) first. The derivative of \( g(x) = x^5 - 3x \) is \( g'(x) = 5x^4 - 3 \). Evaluating \( g'(1) \), we get \( g'(1) = 5(1)^4 - 3 = 2 \).`

So,
`\( h'(4) = (1)/(g'(1)) = (1)/(2) \).`

In summary, the derivative of the inverse function
`\( h \) at \( x = 4 \)` is equal to the reciprocal of the derivative of
`\( g \)` at the corresponding point
`\( x = 1 \), resulting in \( (1)/(2) \).`

Answer 2

To find h'(4), calculate the derivative of g, g'(x), and evaluate it at x=1; then take the reciprocal since g(1)=4. The answer is h'(4) = 1/2.

Step-by-step explanation:

The student is asking for the derivative of the inverse function h at a specific point, given g and g(1)=4. To find h'(4), we need to apply the formula for the derivative of the inverse function, which is 1 over the derivative of the original function at the point x where g(x)=4.

So, we must find g'(x) and then evaluate g'(1) since g(1)=4. The derivative g'(x) is 5x4 - 3. Evaluating the derivative at x=1, we find g'(1)=5(1)4 - 3 = 2. Finally, h'(4) is the reciprocal of g'(1) which gives us h'(4) = 1/2.