AP Calculus AB: 5.1.3 The Derivatives of Trigonometric Functions

• If f ( x ) = sin x , f ′ ( x ) = cos x . If f ( x ) = cos x , f ′ ( x ) = − sin x .

• Use the derivatives of sine and cosine along with different differentiation techniques to find the derivatives of the other trigonometric functions.

• It is not clear what the derivative of the sine function
  is when you apply the formula for the derivative.

• However, you can get a good idea what the graph of
  the derivative looks like by considering the way that
  the slopes of its tangent lines change.

• Notice that the tangent lines start with positive
  slopes. Then the slopes become negative. Then
  the slopes become positive again.

• If you plot the values of the slopes on a graph, you
  will trace out the cosine curve.

• The derivative of the sine function is the cosine
  function.

• The same process can be used on the cosine
  function. However, the results are a little
  unexpected.

• The derivative of the cosine function is the negative
  sine function.

• Find the derivatives of other trigonometric functions
  by expressing them in terms of sine and cosine and
  then applying different computational techniques.

• For example, the tangent function can be expressed
  as a quotient of the sine and cosine functions. So
  finding the derivative of the tangent function
  requires the quotient rule.

• The derivative of the tangent function is the square
  of the secant function.

f ′(t) = 3 sec t (1 + t tan t)

f ′(x) = −2 cot x csc^2 x

f ′(t) = −2 cos t sin t

f′(x)=√2/3 sec^2x

f ′(x) = cos x

f ′(t) = −4 cos^3 t sin t

f ′(t) = 3 tan^2 t sec2 t

f′(x)=3cosx

f ′(t) = 3 sin t (sin t + 2t cos t)

f′(x)=4sec^2x

f′(x)=2tanxsec^2x

f′(x)=−√6cosx

f′(x)=−sin^2x+cos^2x

f′(x)=−1/2sinx

f′(x)=tanxsecx