Exponential, logarithmic and trigonometric derivatives
| English | 中文 | Pinyin |
|---|---|---|
| inner rate | 内函数变化率 | nèi hán shù biàn huà lǜ |
A wave with twice the input frequency has a steeper initial slope. Which factor must appear in its derivative?
- A wave with twice the input frequency has a steeper initial slope. Which factor must appear in its derivative?
- This lesson studies inner rate 内函数变化率: The derivative of the changing input inside a composite function.
Choose the mathematical structure
- For constant k and a>0: (e^(kx))′=k e^(kx), (a^(kx))′=k ln(a)a^(kx), (sin kx)′=k cos kx, (cos kx)′=−k sin kx and (tan kx)′=k sec²(kx), where sec u=1/cos u. Trigonometric inputs are radians. (ln x)′=1/x for x>0. Add differentiated terms and retain their coefficients.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines inner rate?
The derivative of the changing input inside a composite function.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For x>0 and cos(x/2)≠0, let f=2e^(3x)+3^(2x)−4lnx+sin2x−2cos3x+tan(x/2). Then f′=6e^(3x)+2ln3·3^(2x)−4/x+2cos2x+6sin3x+(1/2)sec²(x/2). The term −2cos3x gives a positive 6sin3x because its coefficient and derivative sign are both negative. For g=5^(−2x), g′=−2ln5·5^(−2x), so g′(0)=−2ln5. For h=sin2x−2cos3x+tan(x/2), h′(0)=2+0+1/2=5/2. For p=2e^(3x), p′(0)=6. Base a=1 gives the constant 1 and derivative zero; k=0 also produces constant functions.
Exponential, logarithmic and trigonometric derivatives
For constant k and a>0: (e^(kx))′=k e^(kx), (a^(kx))′=k ln(a)a^(kx), (sin kx)′=k cos kx, (cos kx)′=−k sin kx and (tan kx)′=k sec²(kx), where sec u=1/cos u
Choose the derivative rule and preserve every coefficient, inner rate and input restriction.
For p=2e^(3x), find p′(0).
Retain the coefficient 2 and inner rate 3: 6e⁰=6.
Test a tempting shortcut
- The exponential base is constant but its exponent changes: a^(kx) needs ln a as well as k. The derivative of ln x is 1/x, not ln x/x. Tangent is undefined where cos(kx)=0 and its derivative retains those exclusions. Derivative formulas in radians need π/180 factors for degree variables. A negative k changes the inner-rate sign; it must not disappear.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The derivative of a^(kx) is k a^(kx) for every positive base a. This claim is false. Explain which definition or assumption it violates.
For the worked h, find h′(0).
2cos0+6sin0+(1/2)sec²0=5/2.
The derivative of a^(kx) is k a^(kx) for every positive base a.
The exponential base is constant but its exponent changes: a^(kx) needs ln a as well as k. The derivative of ln x is 1/x, not ln x/x. Tangent is undefined where cos(kx)=0 and its derivative retains those exclusions. Derivative formulas in radians need π/180 factors for degree variables. A negative k changes the inner-rate sign; it must not disappear.
Interpret a new situation
- Identify the function family, preserve its coefficient, differentiate the outer function and multiply by k. List log and tangent domain exclusions before evaluating at a point. Check the sign against an increasing/declining exponential or the local wave direction. Derivatives of linear combinations are obtained by addition, not by multiplying terms.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For y=ln x, find its derivative at x=4.
The positive input 4 gives 1/4.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · G. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The derivative of the changing input inside a composite function. Choose the relationship, show the method, check its assumptions and interpret the result.