What is the derivative of $ln(x^2)+5^(2x)$ ?

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Answer 1 · 2016-10-14T22:09:31

$dy/dx = 2/x + (2ln(5))5^(2x)$

Term-by-term

Term 1:

Before you differentiate use the identity $ln(x^a) = aln(x)$

$2(d[ln(x)])/dx = 2/x$

Term 2:

Use logarithmic differentiation:

$y = 5^(2x)$

$ln(y) = ln(5)2x$

$1/ydy/dx = 2ln(5)$

$dy/dx = (2ln(5))5^(2x)$

Put the two terms back together:

$dy/dx = 2/x + (2ln(5))5^(2x)$

What is the derivative of ln(x^2)+5^(2x)ln(x^2)+5^(2x)?