sin(3*x + 1)*tan(x + 3)
sin(3*x + 1)*tan(x + 3)
Apply the product rule:
; to find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Differentiate term by term:
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 derivative of the constant is zero.
The result is:
The result of the chain rule is:
; to find :
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 :
Differentiate term by term:
Apply the power rule: goes to
The derivative of the constant is zero.
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 :
Differentiate term by term:
Apply the power rule: goes to
The derivative of the constant is zero.
The result is:
The result of the chain rule is:
Now plug in to the quotient rule:
The result is:
Now simplify:
The answer is:
/ 2 \ \1 + tan (x + 3)/*sin(3*x + 1) + 3*cos(3*x + 1)*tan(x + 3)
/ 2 \ / 2 \ -9*sin(1 + 3*x)*tan(3 + x) + 6*\1 + tan (3 + x)/*cos(1 + 3*x) + 2*\1 + tan (3 + x)/*sin(1 + 3*x)*tan(3 + x)
/ 2 \ / 2 \ / 2 \ / 2 \ - 27*\1 + tan (3 + x)/*sin(1 + 3*x) - 27*cos(1 + 3*x)*tan(3 + x) + 2*\1 + tan (3 + x)/*\1 + 3*tan (3 + x)/*sin(1 + 3*x) + 18*\1 + tan (3 + x)/*cos(1 + 3*x)*tan(3 + x)