Question
Find the derivative of the function. \[ y=8^{4 x-9} \] \( y^{\prime}=\square \)
Ask by French Stephens. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( y' = 4 \ln(8) \cdot 8^{4x - 9} \)
Solution
To differentiate y = 8^(4x − 9), use the formula for differentiating an exponential function with a constant base:
d/dx [a^(g(x))] = a^(g(x)) · ln(a) · g′(x).
Here, a = 8 and g(x) = 4x − 9. Compute g′(x):
g′(x) = 4.
Thus,
y′ = 8^(4x − 9) · ln(8) · 4 = 4 ln(8) · 8^(4x − 9).
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Simplify this solution Bonus Knowledge
To find the derivative of the function \( y = 8^{4x - 9} \), we can use the chain rule and the property of exponential functions. The derivative of \( a^u \) is given by \( a^u \ln(a) \cdot u' \), where \( u \) is a function of \( x \). Here, let \( u = 4x - 9 \). Then \( u' = 4 \). Now, applying the derivative formula: \[ y' = 8^{4x - 9} \ln(8) \cdot (4) \] Thus, we can write: \[ y' = 4 \cdot 8^{4x - 9} \ln(8) \] Therefore, the derivative is: \[ y' = 4 \cdot 8^{4x - 9} \ln(8). \]
