Problem to solve
Derivate \[\frac{e^{6^x}}{\sqrt[4]{\sec^3[x+5]}}\]
Solution
STEP 1
Possible derivation: d/dx(e^(6^x)/(sec^(3/4)(x + 5)))
STEP 2
Use the product rule, d/dx(u v) = v ( du)/( dx) + u ( dv)/( dx), where u = e^(6^x) and v = 1/(sec^(3/4)(x + 5)): = e^(6^x) (d/dx(1/(sec^(3/4)(5 + x)))) + (d/dx(e^(6^x)))/(sec^(3/4)(5 + x))
STEP 3
Using the chain rule, d/dx(1/(sec^(3/4)(x + 5))) = d/( du)1/u^(3/4) ( du)/( dx), where u = sec(x + 5) and d/( du)(1/u^(3/4)) = -3/(4 u^(7/4)): = (d/dx(e^(6^x)))/(sec^(3/4)(5 + x)) + (-(3 (d/dx(sec(5 + x))))/(4 sec^(7/4)(5 + x))) e^(6^x)
STEP 4
Using the chain rule, d/dx(sec(x + 5)) = ( dsec(u))/( du) ( du)/( dx), where u = x + 5 and d/( du)(sec(u)) = sec(u) tan(u): = (d/dx(e^(6^x)))/(sec^(3/4)(5 + x)) - ((d/dx(5 + x)) sec(5 + x) tan(5 + x)) (3 e^(6^x))/(4 sec^(7/4)(5 + x))
STEP 5
Simplify the expression: = (d/dx(e^(6^x)))/(sec^(3/4)(5 + x)) - 3/4 e^(6^x) (d/dx(5 + x)) sec(5 + x)^(1/4) sin(5 + x)
STEP 6
Differentiate the sum term by term: = (d/dx(e^(6^x)))/(sec^(3/4)(5 + x)) - d/dx(5) + d/dx(x) 3/4 e^(6^x) sec(5 + x)^(1/4) sin(5 + x)
STEP 7
Using the chain rule, d/dx(e^(6^x)) = ( d e^u)/( du) ( du)/( dx), where u = 6^x and d/( du)(e^u) = e^u: = -3/4 e^(6^x) (d/dx(5) + d/dx(x)) sec(5 + x)^(1/4) sin(5 + x) + e^(6^x) (d/dx(6^x)) 1/(sec^(3/4)(5 + x))
STEP 8
Using the chain rule, d/dx(6^x) = ( d6^u)/( du) ( du)/( dx), where u = x and d/( du)(6^u) = 6^u log(6): = -3/4 e^(6^x) (d/dx(5) + d/dx(x)) sec(5 + x)^(1/4) sin(5 + x) + 6^x log(6) (d/dx(x)) e^(6^x)/(sec^(3/4)(5 + x))
STEP 9
The derivative of x is 1: = -3/4 e^(6^x) (d/dx(5) + d/dx(x)) sec(5 + x)^(1/4) sin(5 + x) + 1 (6^x e^(6^x) log(6))/(sec^(3/4)(5 + x))
STEP 10
The derivative of 5 is zero: = (6^x e^(6^x) log(6))/(sec^(3/4)(5 + x)) - 3/4 e^(6^x) sec(5 + x)^(1/4) sin(5 + x) (d/dx(x) + 0)
STEP 11
Simplify the expression: = (6^x e^(6^x) log(6))/(sec^(3/4)(5 + x)) - 3/4 e^(6^x) (d/dx(x)) sec(5 + x)^(1/4) sin(5 + x)
STEP 12
The derivative of x is 1: Answer: | | = (6^x e^(6^x) log(6))/(sec^(3/4)(5 + x)) - 1 3/4 e^(6^x) sec(5 + x)^(1/4) sin(5 + x)
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Edited by tododemate® - 2021