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Popular Calculus >

derivative of ln((sqrt(x+3)/(e^{3x)}))

Solution

\frac{d}{d x} (ln (\frac{x + 3}{e ^{3 x}}))

Solution

\frac{1}{2 ( x + 3 )} - 3

Solution steps

\frac{d}{d x} (ln (\frac{x + 3}{e ^{3 x}}))

Simplify

ln (\frac{x + 3}{e ^{3 x}}) : ln (x + 3) - 3 x

= \frac{d}{d x} (ln (x + 3) - 3 x)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (ln (x + 3)) - \frac{d}{d x} (3 x)

\frac{d}{d x} (ln (x + 3)) = \frac{1}{2 ( x + 3 )}

\frac{d}{d x} (3 x) = 3

= \frac{1}{2 ( x + 3 )} - 3

Graph

Popular Examples

(\partial)/(\partial x)(x/(9+y))

\frac{\partial}{\partial x} (\frac{x}{9 + y})

derivative of y=sqrt(7x^3-6x^2-8/x)

d er i v a t i v e y = 7 x^{3} - 6 x^{2} - \frac{8}{x}

4y^{''}-y^'+y=0

4 y^{^{''}} - y^{^{'}} + y = 0

y^{''}+2y^'-y=10

y^{^{''}} + 2 y^{^{'}} - y = 10

(dy)/(dx)=sqrt(1-9y^2),y(-1)=0

\frac{d y}{d x} = 1 - 9 y^{2}, y (- 1) = 0

Frequently Asked Questions (FAQ)

What is the derivative of ln((sqrt(x+3)/(e^{3x)})) ?
The derivative of ln((sqrt(x+3)/(e^{3x)})) is 1/(2(x+3))-3
What is the first derivative of ln((sqrt(x+3)/(e^{3x)})) ?
The first derivative of ln((sqrt(x+3)/(e^{3x)})) is 1/(2(x+3))-3