What is derivative of (sinx)^tanhx? If you help me, i am very grateful thanks...

derivative, hyperbolic

1 Answer
Feb 19, 2018

#sin(x)^tanh(x)*(1-tanh^2(x))*ln(sin(x))+"#
#" "sin(x)^(tanh(x)-1)*tanh(x)*cos(x)#

Explanation:

#"The derivative of "#
#f(x)^g(x)#
#"is a difficult formula to remember."#
#"If you can't remember it well, you can deduce it as follows :"#

#x^y = exp(y*ln(x))#
#=> f(x)^g(x) = exp(g(x)*ln(f(x)))#
#=> (f(x)^g(x))' = exp(g(x)*ln(f(x))) (g(x)*ln(f(x)))'#
#"(chain rule + derivative of exp(x))"#
#= exp(g(x)*ln(f(x))) (g'(x)*ln(f(x)) + g(x)(f'(x))/f(x))#
#= f(x)^g(x)*g'(x)*ln(f(x)) + f(x)^(g(x) - 1)*g(x)*f'(x)#

#"Here we have"#
#f(x) = sin(x) => f'(x) = cos(x)#
#g(x) = tanh(x) => g'(x) = 1 - tanh^2(x)#

#= sin(x)^tanh(x)*(1-tanh^2(x))*ln(sin(x)) +"#
#sin(x)^(tanh(x)-1)*tanh(x)*cos(x)#