Here are the derivatives of the given functions using first principles:
1) f(x)=cosx
Step 1: Write the definition of the derivative from first principles.
f′(x)=limh→0hf(x+h)−f(x)
Step 2: Substitute f(x)=cosx and f(x+h)=cos(x+h) into the formula.
f′(x)=limh→0hcos(x+h)−cosx
Step 3: Use the sum-to-product identity cosA−cosB=−2sin(2A+B)sin(2A−B).
Here, A=x+h and B=x.
f′(x)=limh→0h−2sin(2x+h+x)sin(2x+h−x)
f′(x)=limh→0h−2sin(x+2h)sin(2h)
Step 4: Rearrange the terms to use the standard limit limθ→0θsinθ=1.
f′(x)=limh→0(−sin(x+2h)⋅2hsin(2h))
Step 5: Apply the limit as h→0.
f′(x)=−sin(x+0)⋅1
f′(x)=−sinx
2) f(x)=tanx
Step 1: Write the definition of the derivative from first principles.
f′(x)=limh→0hf(x+h)−f(x)
Step 2: Substitute f(x)=tanx and f(x+h)=tan(x+h) into the formula.
f′(x)=limh→0htan(x+h)−tanx
Step 3: Rewrite tanx as cosxsinx and combine the fractions in the numerator.
f′(x)=limh→0hcos(x+h)sin(x+h)−cosxsinx
f′(x)=limh→0hcos(x+h)cosxsin(x+h)cosx−cos(x+h)sinx
Step 4: Use the sine subtraction formula sin(A−B)=sinAcosB−cosAsinB.
Here, A=x+h and B=x.
f′(x)=limh→0hcos(x+h)cosxsin((x+h)−x)
f′(x)=limh→0hcos(x+h)cosxsinh
Step 5: Separate the limit into known parts.
f′(x)=limh→0(hsinh⋅cos(x+h)cosx1)
Step 6: Apply the limit as h→0.
f′(x)=1⋅cos(x+0)cosx1
f′(x)=cosxcosx1
f′(x)=cos2x1
f′(x)=sec2x
That's 2 down. 3 left today — send the next one.
