Derivatives of Exponential and Trigonometric Functions

Learn exponential derivatives, trigonometric derivatives.

Lesson 3 of 32 parts Practice Flashcards

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Exponential Derivatives

Objective: Explain exponential derivatives, apply the stated method, and check its conditions.

The natural exponential satisfies d(eˣ)/dx=eˣ. For e^{g(x)}, use e^{g(x)}g′(x); a constant prefactor remains. For bases other than e, an additional ln(base) factor appears, assuming a positive fixed base. A general base b adds a factor ln b.

Formulas

$d(Ae^{kx})/dx=Ak e^{kx}$

Worked examples

Try itFor f(x)=6e^(7x), find the coefficient multiplying e^(7x) in f′(x).Show answer

Step 1: The derivative of e^(kx) is k e^(kx). Step 2: Multiply coefficient 6 by inner derivative 7. Step 3: f′(x)=42e^(7x).

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Exponential Derivatives

Objective: Explain exponential derivatives, apply the stated method, and check its conditions.

Formulas

$d(Ae^{kx})/dx=Ak e^{kx}$

Worked examples

Try itFor f(x)=6e^(7x), find the coefficient multiplying e^(7x) in f′(x).Show answer

Step 1: The derivative of e^(kx) is k e^(kx). Step 2: Multiply coefficient 6 by inner derivative 7. Step 3: f′(x)=42e^(7x).

Trigonometric Derivatives

Objective: Explain trigonometric derivatives, apply the stated method, and check its conditions.

With angles in radians, d(sin x)/dx=cos x and d(cos x)/dx=−sin x. A scaled input requires a chain-rule factor. Degree-based differentiation has additional conversion factors, so radian assumptions are essential. The standard sine and cosine derivative rules assume radians.

Formulas

$d(A sin(kx))/dx=Ak cos(kx)$

Worked examples

Try itFor f(x)=6sin(7x), with x in radians, find f′(0).Show answer

Step 1: f′(x)=42cos(7x). Step 2: At x=0, cosine is 1. Step 3: f′(0)=42.

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