Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find k^(')(x) if k(x)=-3x*e^(-4x^(3)+3x^(2)).

Find k′(x) k^{\prime}(x) if k(x)=−3x⋅e−4x3+3x2 k(x)=-3 x \cdot e^{-4 x^{3}+3 x^{2}} .

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Write a formula for an arithmetic sequenceright chevron icon

Full solution

Q. Find k′(x) k^{\prime}(x) if k(x)=−3x⋅e−4x3+3x2 k(x)=-3 x \cdot e^{-4 x^{3}+3 x^{2}} .
  1. Identify Components: Identify the components of the function that will require the use of the product rule and the chain rule for differentiation. The function k(x)=−3x⋅e(−4x3+3x2)k(x) = -3x \cdot e^{(-4x^3 + 3x^2)} is a product of two functions, −3x-3x and e(−4x3+3x2)e^{(-4x^3 + 3x^2)}. The exponential function also contains a composite function, which will require the use of the chain rule.
  2. Apply Product Rule: Apply the product rule for differentiation, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Let u=−3xu = -3x and v=e(−4x3+3x2)v = e^{(-4x^3 + 3x^2)}. Then, k′(x)=u′v+uv′k'(x) = u'v + uv'.
  3. Differentiate uu: Differentiate u=−3xu = -3x with respect to xx to get u′=−3u' = -3.
  4. Differentiate vv: Differentiate v=e(−4x3+3x2)v = e^{(-4x^3 + 3x^2)} with respect to xx using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Let g(x)=−4x3+3x2g(x) = -4x^3 + 3x^2 be the inner function. Then, v′=eg(x)⋅g′(x)v' = e^{g(x)} \cdot g'(x).
  5. Differentiate gg: Differentiate g(x)=−4x3+3x2g(x) = -4x^3 + 3x^2 with respect to xx to get g′(x)=−12x2+6xg'(x) = -12x^2 + 6x.
  6. Substitute g′g' into v′v': Substitute g′(x)g'(x) into the expression for v′v' to get v′=e(−4x3+3x2)⋅(−12x2+6x)v' = e^{(-4x^3 + 3x^2)} \cdot (-12x^2 + 6x).
  7. Substitute into Product Rule: Substitute u′u', vv, and v′v' into the product rule expression k′(x)=u′v+uv′k'(x) = u'v + uv' to get k′(x)=−3⋅e(−4x3+3x2)+(−3x)⋅(e(−4x3+3x2)⋅(−12x2+6x))k'(x) = -3 \cdot e^{(-4x^3 + 3x^2)} + (-3x) \cdot (e^{(-4x^3 + 3x^2)} \cdot (-12x^2 + 6x)).
  8. Simplify Expression: Simplify the expression for k′(x)k'(x) by distributing and combining like terms. k′(x)=−3e(−4x3+3x2)−3x⋅e(−4x3+3x2)⋅(−12x2+6x).k'(x) = -3e^{(-4x^3 + 3x^2)} - 3x \cdot e^{(-4x^3 + 3x^2)} \cdot (-12x^2 + 6x).
  9. Factor out Common Factor: Factor out the common factor e(−4x3+3x2)e^{(-4x^3 + 3x^2)} to get k′(x)=e(−4x3+3x2)×(−3−3x×(−12x2+6x))k'(x) = e^{(-4x^3 + 3x^2)} \times (-3 - 3x \times (-12x^2 + 6x)).
  10. Distribute −3x-3x: Continue simplifying the expression by distributing −3x-3x inside the parentheses. k′(x)=e(−4x3+3x2)⋅(−3+36x3−18x2)k'(x) = e^{(-4x^3 + 3x^2)} \cdot (-3 + 36x^3 - 18x^2).
  11. Combine Like Terms: Combine like terms inside the parentheses to get the final expression for k′(x)k'(x). k′(x)=e(−4x3+3x2)⋅(36x3−18x2−3)k'(x) = e^{(-4x^3 + 3x^2)} \cdot (36x^3 - 18x^2 - 3).

More problems from Write a formula for an arithmetic sequence

Question
Find the sum of the finite arithmetic series. ∑n=110(7n+4)\sum_{n=1}^{10} (7n+4)\newline______
Get tutor helpright-arrow

Posted 3 months ago

Question
Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`
Get tutor helpright-arrow

Posted 2 months ago

Question
Type the missing number in this sequence: `2, 4 8`, ______,`32`
Get tutor helpright-arrow

Posted 2 months ago

Question
What kind of sequence is this? 2,10,50,250,…2, 10, 50, 250, \ldots Choices:Choices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}
Get tutor helpright-arrow

Posted 2 months ago

Question
What is the missing number in this pattern? 1,4,9,16,25,36,49,64,81,____1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_
Get tutor helpright-arrow

Posted 2 months ago

Question
Classify the series. ∑n=012(n+2)3 \sum_{n = 0}^{12} (n + 2)^3 \newlineChoices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the first three partial sums of the series.\newline1+6+11+16+21+26+⋯1 + 6 + 11 + 16 + 21 + 26 + \cdots\newlineWrite your answers as integers or fractions in simplest form.\newlineS1=S_1 = ____\newlineS2=S_2 = ____\newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the third partial sum of the series. \newline3+9+15+21+27+33+⋯3 + 9 + 15 + 21 + 27 + 33 + \cdots\newlineWrite your answer as an integer or a fraction in simplest form. \newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the first three partial sums of the series. \newline1+7+13+19+25+31+⋯1 + 7 + 13 + 19 + 25 + 31 + \cdots\newlineWrite your answers as integers or fractions in simplest form. \newlineS1=S_1 = ____\newlineS2=S_2 = ____\newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 3 months ago

Question
Does the infinite geometric series converge or diverge?\newline1+34+916+2764+⋯1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots\newlineChoices:\newline[A]converge\text{[A]converge}\newline[B]diverge\text{[B]diverge}
Get tutor helpright-arrow

Posted 2 months ago

View all (95)