Questions: Let h(x) = tan(5x+3). Then h'(4) is □ and h''(4) is □

Let h(x) = tan(5x+3). Then h'(4) is □ and h''(4) is □
Transcript text: Let $h(x)=\tan (5 x+3)$. Then $h^{\prime}(4)$ is $\square$ and $h^{\prime \prime}(4)$ is $\square$
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Solution

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Solution Steps

To find \( h'(4) \) and \( h''(4) \) for the function \( h(x) = \tan(5x + 3) \), we need to follow these steps:

  1. First Derivative: Use the chain rule to differentiate \( h(x) \) with respect to \( x \).
  2. Evaluate First Derivative: Substitute \( x = 4 \) into the first derivative to find \( h'(4) \).
  3. Second Derivative: Differentiate the first derivative to find the second derivative.
  4. Evaluate Second Derivative: Substitute \( x = 4 \) into the second derivative to find \( h''(4) \).
Step 1: First Derivative

To find the first derivative of the function \( h(x) = \tan(5x + 3) \), we apply the chain rule: \[ h'(x) = 5 \sec^2(5x + 3) \] This can also be expressed as: \[ h'(x) = 5 \tan(5x + 3)^2 + 5 \]

Step 2: Evaluate First Derivative at \( x = 4 \)

Substituting \( x = 4 \) into the first derivative: \[ h'(4) = 5 + 5 \tan(23)^2 \]

Step 3: Second Derivative

Next, we differentiate the first derivative to find the second derivative: \[ h''(x) = 5 \cdot 10 \tan(5x + 3)^2 \sec^2(5x + 3) \] This simplifies to: \[ h''(x) = 5(10 \tan(5x + 3)^2 + 10) \tan(5x + 3) \]

Step 4: Evaluate Second Derivative at \( x = 4 \)

Substituting \( x = 4 \) into the second derivative: \[ h''(4) = 5(10 + 10 \tan(23)^2) \tan(23) \]

Final Answer

Thus, the values are: \[ h'(4) = 5 + 5 \tan(23)^2 \] \[ h''(4) = 5(10 + 10 \tan(23)^2) \tan(23) \] The final boxed answers are: \[ \boxed{h'(4) = 5 + 5 \tan(23)^2} \] \[ \boxed{h''(4) = 5(10 + 10 \tan(23)^2) \tan(23)} \]

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