How do you differentiate $s (x) = {log}_{3} (x^{2} + 5 x)$ $s(x)=log_3(x^2+5x)$ ?

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Answer 1 · 2016-06-14T23:55:19

$s^'(x)=(2x+5)/((x^2+5x)ln(3))$

We can rewrite $s(x)$ using the change of base formula, which states that

$log_a(b)=log_c(b)/log_c(a)$

Here, we will choose a base of $e$ because differentiation with the natural logarithm is simple.

$s(x)=ln(x^2+5x)/ln(3)$

When differentiating this, note that $s(x)$ is really the function $ln(x^2+5x)$ multiplied by the constant $1/ln(3)$ .

To find the derivative of $ln(x^2+5x)$ , we will use the chain rule.

Since $d/dx(ln(x))=1/x$ , we see that $d/dx(ln(u))=1/u*u^'.$

Thus:

$s^'(x)=1/ln(3)*1/(x^2+5x)*d/dx(x^2+5x)$

$s^'(x)=(2x+5)/((x^2+5x)ln(3))$

How do you differentiate s(x)=log_3(x^2+5x)s(x)=log3(x2+5x)s(x)=log_3(x^2+5x)?