Find the Derivative - d/dx y = log base 2 of e^(-x)cos(pix)

Problem

d()/d(x)*(log_2)(e(−x)*cos(π*x))

Solution

1. Apply the change of base formula to rewrite the logarithm in terms of the natural logarithm.

(log_2)(u)=ln(u)/ln(2)

2. Simplify the expression using logarithm properties before differentiating.

1/ln(2)*ln(e(−x)*cos(π*x))=1/ln(2)*(ln(e(−x))+ln(cos(π*x)))

3. Further simplify the first term using the property ln(ea)=a

1/ln(2)*(−x+ln(cos(π*x)))

4. Differentiate the expression with respect to x using the sum rule and the chain rule.

d()/d(x)1/ln(2)*(−x+ln(cos(π*x)))=1/ln(2)*((d(−)*x)/d(x)+d(ln(cos(π*x)))/d(x))

5. Apply the chain rule to the second term, where d(ln(u))/d(x)=1/ud(u)/d(x)

d(ln(cos(π*x)))/d(x)=1/cos(π*x)*(−sin(π*x)⋅π)

6. Simplify the trigonometric ratio using the identity sin(θ)/cos(θ)=tan(θ)

(−π*sin(π*x))/cos(π*x)=−π*tan(π*x)

7. Combine the results into the final derivative expression.

1/ln(2)*(−1−π*tan(π*x))

Final Answer

d()/d(x)*(log_2)(e(−x)*cos(π*x))=−(1+π*tan(π*x))/ln(2)

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