What is the derivative of 1/((a*cos(x+b*sin(x)))) ?

The derivative of 1/((a*cos(x+b*sin(x)))) is -(-asin(x)+bcos(x))/((acos(x)+bsin(x))^2)

What is the first derivative of 1/((a*cos(x+b*sin(x)))) ?

The first derivative of 1/((a*cos(x+b*sin(x)))) is -(-asin(x)+bcos(x))/((acos(x)+bsin(x))^2)

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\frac{d}{d x} (\frac{1}{( a \cdot cos ( x ) + b \cdot sin ( x ) )})

- \frac{- a sin ( x ) + b cos ( x )}{( a cos ( x ) + b sin ( x ) ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{1}{( a cos ( x ) + b sin ( x ) )})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= \frac{d}{d x} ((a cos (x) + b sin (x))^{- 1})

Apply the chain rule:

- \frac{1}{( a cos ( x ) + b sin ( x ) ) ^{2}} \frac{d}{d x} (a cos (x) + b sin (x))

= - \frac{1}{( a cos ( x ) + b sin ( x ) ) ^{2}} \frac{d}{d x} (a cos (x) + b sin (x))

\frac{d}{d x} (a cos (x) + b sin (x)) = - a sin (x) + b cos (x)

= - \frac{1}{( a cos ( x ) + b sin ( x ) ) ^{2}} (- a sin (x) + b cos (x))

Simplify

- \frac{1}{( a cos ( x ) + b sin ( x ) ) ^{2}} (- a sin (x) + b cos (x)) : - \frac{- a sin ( x ) + b cos ( x )}{( a cos ( x ) + b sin ( x ) ) ^{2}}

= - \frac{- a sin ( x ) + b cos ( x )}{( a cos ( x ) + b sin ( x ) ) ^{2}}

What is the derivative of 1/((a*cos(x+b*sin(x)))) ?
The derivative of 1/((a*cos(x+b*sin(x)))) is -(-asin(x)+bcos(x))/((acos(x)+bsin(x))^2)
What is the first derivative of 1/((a*cos(x+b*sin(x)))) ?
The first derivative of 1/((a*cos(x+b*sin(x)))) is -(-asin(x)+bcos(x))/((acos(x)+bsin(x))^2)