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2.7 Derivatives of cos x, sin x, eˣ, and ln x

Syllabus
2020
Topic
2.7
Level
—

Learning objectives

—FUN-3.A—Calculate derivatives of familiar functions—Topic 2.7• FUN-3.A Calculate derivatives of familiar functions.• FUN-3.A.4 Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.• Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.—LIM-3.A—Interpret a limit as a definition of a derivative• LIM-3.A Interpret a limit as a definition of a derivative.• LIM-3.A.1 In some cases, recognizing an expression for the definition of the derivative of a function whose derivative is known offers a strategy for determining a limit.• Enduring understanding LIM-3: Reasoning with definitions, theorems, and properties can be used to determine a limit.

Four Familiar Derivative Rules

The functions sin⁡x\sin x, cos⁡x\cos x, exe^x, and ln⁡x\ln x have standard derivatives that can be applied directly. For the trigonometric rules, xx is measured in radians.

Function f(x)f(x) Derivative f′(x)f'(x) Condition
sin⁡x\sin x cos⁡x\cos x xx in radians
cos⁡x\cos x −sin⁡x-\sin x xx in radians
exe^x exe^x all real xx
ln⁡x\ln x 1x\dfrac{1}{x} x>0x>0

If f(x)=3sin⁡x−2ex+ln⁡xf(x)=3\sin x-2e^x+\ln x, differentiate term by term: f′(x)=3cos⁡x−2ex+1x,x>0.f'(x)=3\cos x-2e^x+\frac{1}{x},\qquad x>0. The domain condition comes from the original ln⁡x\ln x term as well as its derivative.

The cosine rule is the one with a negative sign: (cos⁡x)′=−sin⁡x(\cos x)'=-\sin x. These formulas apply directly only when the input is exactly xx; differentiating a composite input such as sin⁡(x2)\sin(x^2) requires a later rule.

Recognize a Limit as a Derivative

A difference-quotient limit can often be evaluated by identifying it as f′(a)f'(a). Match the whole numerator to the change in one function and the denominator to the corresponding change in input.

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}

Identify the underlying function ff, identify the point aa, verify that the numerator and denominator follow the same derivative-definition form, and then replace the limit by the known value f′(a)f'(a).

Consider lim⁡h→0e2+h−e2h.\lim_{h\to0}\frac{e^{2+h}-e^2}{h}. This is the derivative definition for f(x)=exf(x)=e^x at a=2a=2. Since f′(x)=exf'(x)=e^x, the limit equals f′(2)=e2f'(2)=e^2.

Do not match only the denominator. Both function values in the numerator must correspond to the same point change. If the pattern does not match a derivative definition exactly, another limit method may be needed.