derivative of sin(arctan(3x)) — step-by-step solution

Let's differentiate:

\frac{d}{d x} (\sin (\arctan (3 x)))

Apply the differentiation rule:

\frac{d}{d x} (\sin (f)) = \frac{d}{d x} (f) \cdot \cos (f)

with f =

\arctan (3 x)

ⓘ The derivative of sin(x) is cos(x):

\frac{d}{d x} (\sin (\arctan (3 x)))

\frac{d}{d x} (\arctan (3 x)) \cdot \cos (\arctan (3 x))

\frac{d}{d x} (\arctan (3 x)) \cdot \cos (\arctan (3 x))

For real t, arctan(t) lies in (−π/2, π/2), so its cosine is positive and

\cos (\arctan (t)) = \frac{1}{\sqrt{1 + t^{2}}}

ⅰ

\cos (\arctan (3 x))

\frac{1}{\sqrt{1 + {(3 x)}^{2}}}

\frac{d}{d x} (\arctan (3 x)) \cdot \frac{1}{\sqrt{1 + {(3 x)}^{2}}}

Apply the differentiation rule:

\frac{d}{d x} (\arctan (f)) = \frac{d}{d x} (f) \cdot \frac{1}{f^{2} + 1}

with f =

3 x

ⓘ The derivative of arctan(x) is 1/(1 + x^2):

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

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

\frac{\frac{d}{d x} (3 x)}{{(3 x)}^{2} + 1} \cdot \frac{1}{\sqrt{1 + {(3 x)}^{2}}}

ⓘ To multiply fractions, multiply the numerators together and the denominators together:

\frac{\frac{d}{d x} (3 x)}{{(3 x)}^{2} + 1} \cdot \frac{1}{\sqrt{1 + {(3 x)}^{2}}}

\frac{\frac{d}{d x} (3 x) \cdot 1}{({(3 x)}^{2} + 1) \cdot \sqrt{1 + {(3 x)}^{2}}}

\frac{\frac{d}{d x} (3 x)}{({(3 x)}^{2} + 1) \cdot \sqrt{1 + {(3 x)}^{2}}}

■A constant factor stays in front of the derivative, (k*u)' = k*u':

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

3

■Multiplying powers with the same base adds the exponents:

({(3 x)}^{2} + 1) \cdot \sqrt{1 + {(3 x)}^{2}}

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

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

ⓘ To add fractions, put them over the same denominator, then add the numerators:

1 + \frac{1}{2}

\frac{2 + 1}{2}

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

ⓘ The numerator and the denominator are computed before the division:

2 + 1

3

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

ⓘ The power of a product is the product of the powers:

{(3 x)}^{2}

3^{2} \cdot x^{2}

\frac{3}{{(3^{2} \cdot x^{2} + 1)}^{\frac{3}{2}}}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

3^{2}

3 \cdot 3

9

▶Derivative:

\frac{3}{{(9 x^{2} + 1)}^{\frac{3}{2}}}