Final answer to the problem
$\frac{1}{2}\left(-3\sin\left(3x\right)\ln\left(1+e^{5x}\right)+\frac{5e^{5x}\cos\left(3x\right)}{1+e^{5x}}\right)\left(1+e^{5x}\right)^{\frac{1}{2}\cos\left(3x\right)}$
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Step-by-step Solution
Learn how to solve sum rule of differentiation problems step by step online. Find the derivative using logarithmic differentiation method d/dx((1+e^(5x))^(1/2)^cos(3x)). Simplify the derivative by applying the properties of logarithms. To derive the function \left(1+e^{5x}\right)^{\frac{1}{2}\cos\left(3x\right)}, use the method of logarithmic differentiation. First, assign the function to y, then take the natural logarithm of both sides of the equation. Apply natural logarithm to both sides of the equality. Using the power rule of logarithms: \log_a(x^n)=n\cdot\log_a(x).
Final answer to the problem
$\frac{1}{2}\left(-3\sin\left(3x\right)\ln\left(1+e^{5x}\right)+\frac{5e^{5x}\cos\left(3x\right)}{1+e^{5x}}\right)\left(1+e^{5x}\right)^{\frac{1}{2}\cos\left(3x\right)}$
