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Let g(x)=x^(5)+3x and let h be the inverse function of g. Notice that g(1)=4. h^(')(4)=

Let g(x)=x5+3x g(x)=x^{5}+3 x and let h h be the inverse function of g g . Notice that g(1)=4 g(1)=4 .\newlineh′(4)= h^{\prime}(4)=

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Full solution

Q. Let g(x)=x5+3x g(x)=x^{5}+3 x and let h h be the inverse function of g g . Notice that g(1)=4 g(1)=4 .\newlineh′(4)= h^{\prime}(4)=
  1. Use Inverse Function Derivative Formula: To find h′(4)h'(4), we need to use the formula for the derivative of the inverse function: h′(g(x))=1g′(x)h'(g(x)) = \frac{1}{g'(x)}.
  2. Find g′(x)g'(x): First, we need to find g′(x)g'(x), which is the derivative of g(x)=x5+3xg(x) = x^5 + 3x.\newlineg′(x)=5x4+3g'(x) = 5x^4 + 3.
  3. Calculate g′(1)g'(1): Now we plug in x=1x = 1 into g′(x)g'(x) since g(1)=4g(1) = 4 and we want h′(4)h'(4).\newlineg′(1)=5(1)4+3=5+3=8g'(1) = 5(1)^4 + 3 = 5 + 3 = 8.
  4. Apply Inverse Function Derivative Formula: Now we use the formula for the derivative of the inverse function. h′(4)=1g′(1)=18h'(4) = \frac{1}{g'(1)} = \frac{1}{8}.

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