derivative of (tan^2(x)/(tan(x^2)))

⚠Ⓘ sin^n(u), cos^n(u) and tan^n(u) are read as powers of the function value: write (sin(u))^n, (cos(u))^n or (tan(u))^n

1) Let's differentiate:

\frac{d}{d x} (\frac{{\tan (x)}^{2}}{\tan (x^{2})})

We differentiate

u1

{\tan (x)}^{2}

and

v2

\tan (x^{2})

using:

\frac{d}{d x} (\frac{u1}{v2}) = \frac{\frac{d}{d x} (u1) \cdot v2 - u1 \cdot \frac{d}{d x} (v2)}{{v2}^{2}}

2) Let's perform the differentiation

u1

\frac{d}{d x} ({\tan (x)}^{2})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

with f =

\tan (x)

2 \cdot \frac{d}{d x} (\tan (x)) \cdot \tan (x)

2 \cdot \frac{1}{{\cos (x)}^{2}} \cdot \tan (x)

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}}

3) Let's differentiate

v2

\frac{d}{d x} (\tan (x^{2}))

We differentiate using:

\frac{d}{d x} (\tan (f)) = \frac{d}{d x} (f) \cdot \frac{1}{{\cos (f)}^{2}}

(where f =

x^{2}

)

\frac{d}{d x} (x^{2}) \cdot \frac{1}{{\cos (x^{2})}^{2}}

\frac{2 x}{{\cos (x^{2})}^{2}}

4) Let's evaluate the resulting derivative (

\frac{u1}{v2}

)':

\frac{\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}} \cdot \tan (x^{2}) - {\tan (x)}^{2} \cdot \frac{2 x}{{\cos (x^{2})}^{2}}}{{\tan (x^{2})}^{2}}

\frac{\frac{2 \cdot \tan (x) \cdot \tan (x^{2}) \cdot {\cos (x^{2})}^{2} - {\tan (x)}^{2} \cdot 2 x \cdot {\cos (x)}^{2}}{{\cos (x)}^{2} \cdot {\cos (x^{2})}^{2}}}{{\tan (x^{2})}^{2}}

\frac{\frac{2 \cdot \tan (x) \cdot \tan (x^{2}) \cdot {\cos (x^{2})}^{2} - {\sin (x)}^{2} \cdot 2 x}{{\cos (x)}^{2} \cdot {\cos (x^{2})}^{2}}}{{\tan (x^{2})}^{2}}

\frac{2 \cdot \tan (x) \cdot \tan (x^{2}) \cdot {\cos (x^{2})}^{2} - {\sin (x)}^{2} \cdot 2 x}{{\cos (x)}^{2} \cdot {\cos (x^{2})}^{2}} \cdot \frac{1}{{\tan (x^{2})}^{2}}

\frac{2 \cdot \tan (x) \cdot \tan (x^{2}) \cdot {\cos (x^{2})}^{2} - {\sin (x)}^{2} \cdot 2 x}{{\cos (x)}^{2} \cdot {\cos (x^{2})}^{2} \cdot {\tan (x^{2})}^{2}}

\frac{2 \cdot \tan (x) \cdot \tan (x^{2}) \cdot {\cos (x^{2})}^{2} - {\sin (x)}^{2} \cdot 2 x}{{\cos (x)}^{2} \cdot {\sin (x^{2})}^{2}}

▶Derivative:

\frac{2 \cdot {\cos (x^{2})}^{2} \cdot \tan (x) \cdot \tan (x^{2}) - 2 x \cdot {\sin (x)}^{2}}{{\cos (x)}^{2} \cdot {\sin (x^{2})}^{2}}