Problem to solve
Derivate \[\csc[\tan^2[-3x^5-2]]\]
Solution
STEP 1
Possible derivation: d/dx(csc(tan^2(-3 x^5 - 2)))
STEP 2
Rewrite the expression: csc(tan^2(-3 x^5 - 2)) = csc(tan^2(2 + 3 x^5)): = d/dx(csc(tan^2(2 + 3 x^5)))
STEP 3
Using the chain rule, d/dx(csc(tan^2(3 x^5 + 2))) = ( dcsc(u))/( du) ( du)/( dx), where u = tan^2(3 x^5 + 2) and d/( du)(csc(u)) = -cot(u) csc(u): = -cot(tan^2(2 + 3 x^5)) csc(tan^2(2 + 3 x^5)) (d/dx(tan^2(2 + 3 x^5)))
STEP 4
Using the chain rule, d/dx(tan^2(3 x^5 + 2)) = ( du^2)/( du) ( du)/( dx), where u = tan(3 x^5 + 2) and d/( du)(u^2) = 2 u: = -cot(tan^2(2 + 3 x^5)) csc(tan^2(2 + 3 x^5)) (2 (d/dx(tan(2 + 3 x^5))) tan(2 + 3 x^5))
STEP 5
Simplify the expression: = -2 cot(tan^2(2 + 3 x^5)) csc(tan^2(2 + 3 x^5)) (d/dx(tan(2 + 3 x^5))) tan(2 + 3 x^5)
STEP 6
Using the chain rule, d/dx(tan(3 x^5 + 2)) = ( dtan(u))/( du) ( du)/( dx), where u = 3 x^5 + 2 and d/( du)(tan(u)) = sec^2(u): = -2 cot(tan^2(2 + 3 x^5)) csc(tan^2(2 + 3 x^5)) tan(2 + 3 x^5) ((d/dx(2 + 3 x^5)) sec^2(2 + 3 x^5))
STEP 7
Differentiate the sum term by term and factor out constants: = -2 cot(tan^2(2 + 3 x^5)) csc(tan^2(2 + 3 x^5)) sec^2(2 + 3 x^5) tan(2 + 3 x^5) (d/dx(2) + 3 (d/dx(x^5)))
STEP 8
The derivative of 2 is zero: = -2 cot(tan^2(2 + 3 x^5)) csc(tan^2(2 + 3 x^5)) sec^2(2 + 3 x^5) (3 (d/dx(x^5)) + 0) tan(2 + 3 x^5)
STEP 9
Simplify the expression: = -6 cot(tan^2(2 + 3 x^5)) csc(tan^2(2 + 3 x^5)) (d/dx(x^5)) sec^2(2 + 3 x^5) tan(2 + 3 x^5)
STEP 10
Use the power rule, d/dx(x^n) = n x^(n - 1), where n = 5. d/dx(x^5) = 5 x^4: = -6 cot(tan^2(2 + 3 x^5)) csc(tan^2(2 + 3 x^5)) sec^2(2 + 3 x^5) tan(2 + 3 x^5) 5 x^4
STEP 11
Simplify the expression: Answer: | | = -30 x^4 cot(tan^2(2 + 3 x^5)) csc(tan^2(2 + 3 x^5)) sec^2(2 + 3 x^5) tan(2 + 3 x^5)
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Edited by tododemate® - 2021