Solve cot(θ10^-19)

$$\cot ( \theta { 10 }^{ -19 } )$$

Differentiate w.r.t. θ

$$-\frac{1}{10000000000000000000\left(\sin(\frac{\theta }{10000000000000000000})\right)^{2}}$$

Steps Using Chain Rule

Calculate 10 to the power of -19 and get \frac{1}{10000000000000000000}.

$$\frac{\mathrm{d}}{\mathrm{d}\theta }(\cot(\theta \times \frac{1}{10000000000000000000}))$$

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).

$$\left(-\left(\csc(\frac{1}{10000000000000000000}\theta ^{1})\right)^{2}\right)\frac{\mathrm{d}}{\mathrm{d}\theta }(\frac{1}{10000000000000000000}\theta ^{1})$$

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}.

$$\left(-\left(\csc(\frac{1}{10000000000000000000}\theta ^{1})\right)^{2}\right)\times \frac{1}{10000000000000000000}\theta ^{1-1}$$

Simplify.

$$-\frac{1}{10000000000000000000}\left(\csc(\frac{1}{10000000000000000000}\theta ^{1})\right)^{2}$$

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

$$-\frac{1}{10000000000000000000}\left(\csc(\frac{1}{10000000000000000000}\theta )\right)^{2}$$

Evaluate

$$\cot(\frac{\theta }{10000000000000000000})$$