Solve =tan3(2a)

$$= \tan 3 ( 2 a )$$

Differentiate w.r.t. a

$$\frac{3\left(96\left(\sin(a)\right)^{2}\left(\cos(a)\right)^{10}+96\left(\cos(a)\right)^{2}\left(\sin(a)\right)^{10}+240\left(\sin(a)\right)^{4}\left(\cos(a)\right)^{8}+240\left(\cos(a)\right)^{4}\left(\sin(a)\right)^{8}+16\left(\sin(a)\right)^{12}+16\left(\cos(a)\right)^{12}+5\left(\sin(2a)\right)^{6}\right)}{8\left(\left(-3\sin(a)\left(\cos(a)\right)^{2}-3\cos(a)\left(\sin(a)\right)^{2}+\left(\sin(a)\right)^{3}+\left(\cos(a)\right)^{3}\right)\left(3\sin(a)\left(\cos(a)\right)^{2}-3\cos(a)\left(\sin(a)\right)^{2}+\left(\cos(a)\right)^{3}-\left(\sin(a)\right)^{3}\right)\right)^{2}}$$

Steps Using Chain Rule

Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\tan(6a))$$

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).

$$\left(\sec(6a^{1})\right)^{2}\frac{\mathrm{d}}{\mathrm{d}a}(6a^{1})$$

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}.

$$\left(\sec(6a^{1})\right)^{2}\times 6a^{1-1}$$

Simplify.

$$6\left(\sec(6a^{1})\right)^{2}$$

For any term t, t^{1}=t.

$$6\left(\sec(6a)\right)^{2}$$

Evaluate

$$\tan(6a)$$