Solve tan({left(x/2right)}^2)

$$\tan ( { \left( \frac{ x }{ 2 } \right) }^{ 2 } )$$

Differentiate w.r.t. x

$$\frac{x}{2\left(\cos(\frac{x^{2}}{4})\right)^{2}}$$

Steps Using Chain Rule

To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\tan(\frac{x^{2}}{4}))$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$\left(\sec(\frac{1}{4}x^{2})\right)^{2}\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4}x^{2})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\left(\sec(\frac{1}{4}x^{2})\right)^{2}\times 2\times \frac{1}{4}x^{2-1}$$

Simplify.

$$\frac{x}{2}\left(\sec(\frac{1}{4}x^{2})\right)^{2}$$

Evaluate

$$\tan(\frac{x^{2}}{4})$$