Step-by-step explanation:
f(t) = (1/t²) (csc t − 4)
Rewrite 1/t² using negative exponents:
f(t) = t⁻² (csc t − 4)
To take the derivative, you'll need three derivative rules:
Power rule:
d/dx (xⁿ) = n xⁿ⁻¹
Product rule:
d/dx (f(x) g(x)) = f'(x) g(x) + f(x) g'(x)
Derivative of cosecant:
d/dx (csc x) = -csc x cot x
We'll start by applying product rule:
f'(t) = d/dt (t⁻²) (csc t − 4) + t⁻² d/dt (csc t − 4)
Next we apply power rule:
f'(t) = (-2 t⁻³) (csc t − 4) + t⁻² d/dt (csc t − 4)
Finally, we use derivative of cosecant:
f'(t) = (-2 t⁻³) (csc t − 4) + t⁻² (-csc t cot t)
Simplifying:
f'(t) = -2 t⁻³ (csc t − 4) − t⁻² csc t cot t
f'(t) = -t⁻³ (2 (csc t − 4) + t csc t cot t)
f'(t) = -t⁻³ (2 csc t − 8 + t csc t cot t)
f'(t) = -(2 csc t − 8 + t csc t cot t) / t³
