4*sin(5*x) + 2*tan(5*x)
4*sin(5*x) + 2*tan(5*x)
Differentiate term by term:
The derivative of a constant times a function is the constant times the derivative of the function.
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result of the chain rule is:
So, the result is:
The derivative of a constant times a function is the constant times the derivative of the function.
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result of the chain rule is:
To find :
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result of the chain rule is:
Now plug in to the quotient rule:
So, the result is:
The result is:
Now simplify:
The answer is:
2 10 + 10*tan (5*x) + 20*cos(5*x)
/ / 2 \ \ 100*\-sin(5*x) + \1 + tan (5*x)/*tan(5*x)/
/ 2 \ |/ 2 \ 2 / 2 \| 500*\\1 + tan (5*x)/ - cos(5*x) + 2*tan (5*x)*\1 + tan (5*x)//