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Calculus 1 : How to find differential functions

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Example Question #31 : How To Find Differential Functions

Solve for f{}'(x) when

$f(x)= \left(9x^4+4x^3+\frac{x}{2}\right)^9$

Possible Answers:

$f{}'(x)=9(9x^4+4x^3+\frac{x}{2})^8$

$f{}'(x)=(9x^4+4x^3+\frac{x}{2})^8(324x^3+108x^2+\frac{9}{2})$

$f{}'(x)=(9x^4+4x^3+\frac{x}{2})^9(36x^3+12x^2+\frac{1}{2})$

$f{}'(x)=36x^3+12x^2+\frac{1}{2}$

Correct answer:

$f{}'(x)=(9x^4+4x^3+\frac{x}{2})^8(324x^3+108x^2+\frac{9}{2})$

Explanation:

$f(x)= \left(9x^4+4x^3+\frac{x}{2}\right)^9$

using the chain rule:

$\frac{d}{dx}(f(g(x)))=f{}'(g(x))g{}'(x)$

$f{}'(x)=9(9x^4+4x^3+x/2)^8(36x^3+12x^2+1/2)$

multiply the constant 9 into the second function to simplify answer

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Example Question #32 : How To Find Differential Functions

Solve for f{}'(x) when $f(x)=\log_{9}x^2$

Possible Answers:

$f{}'(x)=\frac{2}{\ln 9}$

$f{}'(x)=\frac{2}{x\ln 9}$

$f{}'(x)=\frac{x}{\ln 9}$

$f{}'(x)= \frac{2}{x}$

Correct answer:

$f{}'(x)=\frac{2}{x\ln 9}$

Explanation:

$f(x)=\log_{9}x^2$

using the logarithm identities change the equation to base 10:

$f(x)=\frac{\ln x^2}{\ln 9}$

using lograthim identities simplify the numerator:

$f(x)=\frac{2\ln x}{\ln 9}$

differentiate

$f{}'(x)=\frac{2*\frac{1}{x}}{\ln 9}$

Simplify

$f{}'(x)=\frac{2}{x\ln 9}$

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Example Question #33 : How To Find Differential Functions

Solve for when

$f(x)= (x^2 +7x)(3x^3+\frac{x}{4})$

Possible Answers:

$f'(x) = 81x^2 + \frac{x}{2}+\frac{7}{4}$

$f{}'(x)= 15x^4 + 21x^3 +255x^2 + 7x +7$

$f^{'}(x)=15x^{4}+84x^{3}+\frac{3x^{2}}{4}+\frac{7x}{2}$

$f{}'(x)= 6x^4 +21x^3 +\frac{x^2}{2}+\frac{7x}{4}$

Correct answer:

$f^{'}(x)=15x^{4}+84x^{3}+\frac{3x^{2}}{4}+\frac{7x}{2}$

Explanation:

$f(x)= (x^2 +7x)(3x^3+\frac{x}{4})$

Using the product rule: (fg){}'=f{}'g+fg{}'

$f{}'(x)= (2x+7)(3x^3 + x/4) + (x^2+7x)(9x^2+1/4)$

FOIL

$f{}'(x)= 6x^4+2x^2/4+21x^3+7x/4+9x^4+x^2/4+63x^3+7x/4$

Combine like-terms

$f{}'(x)= 15x^4 + 84x^3+\frac{3x^2}{4}+\frac{14x}{4}$

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Example Question #34 : How To Find Differential Functions

Solve for y{}' using implicit differentiation if

x^3y^2+sin(y)=11x-x^2y

Possible Answers:

$y{}'=\frac{2x^3y+cos(y)+x^2}{11-2xy-3x^2y^2}$

$y{}'=\frac{11-2xy-3x^2y^2}{2x^3y+cos(y)+x^2}$

$y{}'=-cos(y)+3x^2+2x-11$

$y{}'=-2x+11-cos(y)-6x^2y$

Correct answer:

$y{}'=\frac{11-2xy-3x^2y^2}{2x^3y+cos(y)+x^2}$

Explanation:

x^3y^2+sin(y)=11x-x^2y

Differentiate the equation

$3x^2y^2 + 2x^3y*y{}' + cos(y)*y{}' = 11-(2xy+x^2*y{}')$

Simplify

$3x^2y^2 + 2x^3y*y{}' + cos(y)*y{}' = 11-2xy-x^2*y{}'$

place all terms with y{}' on one side and the other terms on the other side

$2x^3y*y{}'+cos(y)*y{}'+x^2*y{}'=11-2xy-3x^2y^2$

Simplify

$y{}'*(2x^3y+cos(y)+x^2)=11-2xy-3x^2y^2$

Divide and solve for y{}'

$y{}'=\frac{11-2xy-3x^2y^2}{2x^3y+cos(y)+x^2}$

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Example Question #35 : How To Find Differential Functions

Solve for f{}'(c) using the Mean Value Theorem, rounded to the nearest hundredth place when

$f(x)=2x^4 - \frac{x^2}{7}$ on the interval $\left [ 1,3 \right ]$

Possible Answers:

75.24

83.91

79.43

80.65

Correct answer:

79.43

Explanation:

Mean Value Theorem (MVT) = $f{}'(c)= \frac{f(b)-f(a)}{b-a}$ on $\left [ a,b \right ]$

$f{}'(c)= \frac{(2(3)^4-\frac{(3)^2)}{7})- (2(1)^4-\frac{(1)^2}{7}))}{3-1}$

f{}'(c)= 79.43

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Example Question #31 : Other Differential Functions

Evaluate the limit:

$Lim _{x \rightarrow -1} \frac{x^{^{2}}+4x+3}{x+1}$

Possible Answers:

Does Not Exist

Correct answer:

Explanation:

Attempting to evaluate directly (plug in -1 for ) results in the indeterminate form:

$\frac{0}{0}$

Further analysis is required:

$Lim _{x \rightarrow -1} \frac{x^{^{2}}+4x+3}{x+1}\dot{} = Lim _{x \rightarrow -1} \frac{(x+3)(x+1)}{x+1}\dot{} =Lim _{x \rightarrow -1}x+3$

This final form can be evaluated directly:

$Lim _{x \rightarrow -1} x+3 = -1 + 3=2$

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Example Question #37 : How To Find Differential Functions

Evaluate the limit:

$Lim _{x \rightarrow 2} \frac{x^2 +5x-14}{x-2}$

Possible Answers:

Does Not Exist

Correct answer:

Explanation:

Attempting to evaluate directly (plug in 2 for ) results in the indeterminate form:

$\frac{0}{0}$

Further analysis is required:

$Lim _{x \rightarrow 2} \frac{x^2 +5x-14}{x-2}\dot{} = Lim _{x \rightarrow 2} \frac{(x+7)(x-2)}{x-2}\dot{} =Lim _{x \rightarrow 2}x+7$

This final form can be evaluated directly:

$Lim _{x \rightarrow 2} x+7 = 2+7=9$

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Example Question #38 : How To Find Differential Functions

Evaluate the limit:

$Lim _{x \rightarrow 3} \frac{x^2 + 7x +6}{x-3}$

Possible Answers:

Does Not Exist

Correct answer:

Does Not Exist

Explanation:

We evaluate the limit directly (plug in 3 for ) and obtain:

$\frac{36}{0}$

from which we determine that the the function has a vertical asymptote at this point (it goes off to positive or negative infinity). The limit Does Not Exist.

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Example Question #39 : How To Find Differential Functions

Given:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$

$Lim _{x \rightarrow 0} \frac{1 - cosx}{x} = 0$

Evaluate the limit:

$Lim _{x \rightarrow 0} xsecx$

Possible Answers:

$\pi$

Does Not Exist

Correct answer:

Explanation:

This limit can be evaluated directly.

Recall that $secx = \frac{1}{cosx}$

So:

$Lim _{x \rightarrow 0} xsecx = Lim _{x \rightarrow 0} \frac{x}{cosx}= \frac{0}{1}=0$

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Example Question #31 : Other Differential Functions

Given:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$

$Lim _{x \rightarrow 0} \frac{1 - cosx}{x} = 0$

Evaluate the limit:

$Lim _{x \rightarrow 0} xcsc(2\pi x)$

Possible Answers:

$\frac{1}{2}$

$2\pi$

DoesNotExist

$\frac{1}{2\pi}$

Correct answer:

$\frac{1}{2\pi}$

Explanation:

First observe that $xcsc(2\pi x) = \frac{x}{sin2\pi x}$

Multiplying by $\frac{2\pi x}{2\pi x}$ we obtain:

$Lim _{x \rightarrow 0} \frac{2 \pi x*x}{2\pi x*sin2\pi x} = Lim _{x \rightarrow 0} \frac{2\pi x}{sin2\pi x}*\frac{x}{2\pi x}$

Limit of product is the product of limits:

$Lim _{x \rightarrow 0} \frac{2\pi x}{sin2\pi x}*\frac{x}{2\pi x}=Lim _{x \rightarrow 0} \frac{2\pi x}{sin2\pi x}*Lim _{x \rightarrow 0}\frac{x}{2\pi x}$

From the Pre-Question Text:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$

So:

$Lim _{x \rightarrow 0} \frac{2\pi x}{sin2\pi x}*Lim _{x \rightarrow 0}\frac{x}{2\pi x}=1*\frac{1}{2\pi}=\frac{1}{2\pi}$

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Calculus 1 : How to find differential functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #31 : How To Find Differential Functions

Example Question #32 : How To Find Differential Functions

Example Question #33 : How To Find Differential Functions

Solve for when

$f(x)= (x^2 +7x)(3x^3+\frac{x}{4})$

Example Question #34 : How To Find Differential Functions

Example Question #35 : How To Find Differential Functions

Example Question #31 : Other Differential Functions

Example Question #37 : How To Find Differential Functions

Example Question #38 : How To Find Differential Functions

Example Question #39 : How To Find Differential Functions

Example Question #31 : Other Differential Functions

All Calculus 1 Resources

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