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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

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10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #172 : How To Find Differential Functions

Differentiate the function:

$f(x)=\left(\frac{1}{x^2}-\frac{3}{x^4}\right)(x+5x^3)$

Possible Answers:

$(-2x^{-3}+12x^{-5})+(1+15x^2)$

$(x^{-2}-3x^{-4})(1+15x^2)+(x+5x^3)(-2x^{-3}+12x^{-5})$

$\left(\frac{1}{x^2}-\frac{3}{x^4}\right)+(x+5x^2)$

$(-2x^{-3}+12x^{-5})(1+15x^2)$

Correct answer:

$(x^{-2}-3x^{-4})(1+15x^2)+(x+5x^3)(-2x^{-3}+12x^{-5})$

Explanation:

Apply the product rule:

$\frac{d}{dx}[f(x)*g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$

Then apply the product rule $nx^{n-1}$ (where "n" is the exponent) where needed.

$\\ \frac{d}{dx}\left(\left(\frac{1}{x^2}-\frac{3}{x^4}\right)(x+5x^3)\right)\\ \\=(x^{-2}-3x^{-4})\frac{d}{dx}(x+5x^3)+(x+5x^3)\frac{d}{dx}(x^{-2}-3x^{-4})\\ \\=(x^{-2}-3x^{-4})(1*x^{1-1}+3*5x^{3-1})+(x+5x^3)(-2x^{-2-1}-(-4)*3x^{-4-1})\\ \\=(x^{-2}-3x^{-4})(1+15x^2)+(x+5x^3)(-2x^{-3}+12x^{-5})$

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Example Question #1391 : Calculus

Differentiate the following function:

$f(x)=(x^3-2x)(x^{-4}+x^{-2})$

Possible Answers:

$(x^3-2x)+(x^{-4}+x^{-2})$

$(-4x^{-5}-2x^{-3})+(3x^2-2)$

$(-4x^{-5}-2x^{-3})(3x^2-2)$

$(x^3-2x)(-4x^{-5}-2x^{-3})+(x^{-4}+x^{-2})(3x^2-2)$

Correct answer:

$(x^3-2x)(-4x^{-5}-2x^{-3})+(x^{-4}+x^{-2})(3x^2-2)$

Explanation:

Apply the product rule:

$\frac{d}{dx}[f(x)*g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$

Then apply the product rule $nx^{n-1}$ (where "n" is the exponent) where needed.

$\\ \frac{d}{dx}[(x^3-2x)(x^{-4}+x^{-2})]\\ \\=(x^3-2x)\frac{d}{dx}(x^{-4}+x^{-2})+(x^{-4}+x^{-2})\frac{d}{dx}(x^3-2x)\\ \\=(x^3-2x)((-4)x^{-4-1}+(-2)x^{-2-1})+(x^{-4}+x^{-2})(3x^{3-1}-2x^{1-1})\\ \\=(x^3-2x)(-4x^{-5}-2x^{-3})+(x^{-4}+x^{-2})(3x^2-2)$

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Example Question #1391 : Calculus

Differentiate the function:

f(x)=sin(x)+cot(x)

Possible Answers:

cos(x)-csc^2(x)

cos(x)+csc^2(x)

cos(x)-csc(x)

-cos(x)-csc^2(x)

Correct answer:

cos(x)-csc^2(x)

Explanation:

Because this is a sum of functions, take the derivate of each function with respect to "x" and add them together:

$\\ \frac{d}{dx}[sin(x)+cot(x)]\\ \\=\frac{d}{dx}(sin(x))+\frac{d}{dx}(cot(x))\\ \\=cos(x)+(-csc^(x))\\ \\=cos(x)-csc^2(x)$

The derivative of sin(x) is cos(x) and the derivate of cot(x) is -csc^2(x) .

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Example Question #174 : How To Find Differential Functions

Differentiate the function:

f(x)=x^2sin(x)

Possible Answers:

$2x\cos (x)$

$x^2\cos (x)+2x\sin(x)$

$2x+\cos (x)$

Correct answer:

$x^2\cos (x)+2x\sin(x)$

Explanation:

Apply the product rule:

$\frac{d}{dx}[f(x)*g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$

Then apply the product rule $nx^{n-1}$ (where "n" is the exponent) where needed.

$\\ \frac{d}{dx}[x^2sin(x)]\\ \\=x^2\frac{d}{dx}(sin(x))+sin(x)\frac{d}{dx}(x^2)\\ \\=x^2cos(x)+sin(x)2x^{2-1}\\ \\=x^2cos(x)+2xsin(x)$

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Example Question #175 : How To Find Differential Functions

Differentiate the function:

$f(x)=(x^2+2)(x^{\frac{1}{2}}-1)$

Possible Answers:

$(x^2+2)+(x^{\frac{1}{2}}-1)$

$2x+x^{-\frac{1}{2}}$

$\frac{5}{2}x^{\frac{3}{2}}-2x+x^{-\frac{1}{2}}$

$2x^{\frac{1}{2}}$

Correct answer:

$\frac{5}{2}x^{\frac{3}{2}}-2x+x^{-\frac{1}{2}}$

Explanation:

Apply the product rule:

$\frac{d}{dx}[f(x)*g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$

Then apply the product rule $nx^{n-1}$ (where "n" is the exponent) where needed.

$\\ \frac{d}{dx}[(x^2+2)(x^{\frac{1}{2}}-1)]\\ \\=(x^2+2)\frac{d}{dx}(x^{\frac{1}{2}}-1)+(x^{\frac{1}{2}}-1)\frac{d}{dx}(x^2+2)\\ \\=(x^2+2)(\frac{1}{2}x^{\frac{1}{2}-1}-0)+(x^{\frac{1}{2}}-1)(2x^{2-1}+0)\\ \\=\frac{1}{2}x^{-\frac{1}{2}}(x^2+2)+2x(x^{\frac{1}{2}}-1)\\ \\=\frac{1}{2}x^{\frac{3}{2}}+x^{-\frac{1}{2}}+2x^{\frac{3}{2}}-2x\\ \\=\frac{5}{2}x^{\frac{3}{2}}-2x+x^{-\frac{1}{2}}$

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Example Question #1392 : Calculus

Differentiate the function:

$f(x)=\sin (x)\cos (x)$

Possible Answers:

$\cos (x)-\sin (x)$

$-\sin (x)\cos (x)$

$\cos (x)+\sin (x)$

$\cos^2(x)-\sin^2(x)$

Correct answer:

$\cos^2(x)-\sin^2(x)$

Explanation:

Apply the product rule:

$\frac{d}{dx}[f(x)*g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$

Let f(x)=sin(x) and g(x)=cos(x)

$\\ \frac{d}{dx}[\sin (x)\cos (x)]\\ \\=\sin (x)\frac{d}{dx}\cos (x)+\cos (x)\frac{d}{dx}\sin (x)\\ \\=-\sin (x)\sin (x)+\cos (x)\cos (x)\\ \\=\cos^2(x)-\sin^2(x)$

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Example Question #1391 : Calculus

Differentiate the function:

$f(x)=(x^3+5x)(x^{-2}-6x^2)$

Possible Answers:

$(x^3+5x)+(x^{-2}-6x^2)$

$(x^3+5x)(-2x^{-3}-12x)+(x^{-2}-6x^2)(3x^2+5)$

$(-2x^{-3}-12x)(3x^2+5)$

$(-2x^{-3}-12x)+(3x^2+5)$

Correct answer:

$(x^3+5x)(-2x^{-3}-12x)+(x^{-2}-6x^2)(3x^2+5)$

Explanation:

Apply the product rule:

$\frac{d}{dx}[f(x)*g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$

Then apply the product rule $nx^{n-1}$ (where "n" is the exponent) where needed.

$\\ \frac{d}{dx}[(x^3+5x)(x^{-2}-6x^2)]\\ \\=(x^3+5x)\frac{d}{dx}(x^{-2}-6x^2)+(x^{-2}-6x^2)\frac{d}{dx}(x^3+5x)\\ \\=(x^3+5x)(-2x^{-2-1}-(2)6x^{2-1})+(x^{-2}-6x^2)(3x^{3-1}+5x^{1-1})\\ \\=(x^3+5x)(-2x^{-3}-12x)+(x^{-2}-6x^2)(3x^2+5)$

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Example Question #182 : How To Find Differential Functions

Differentiate the function:

$f(x)=(x^2+x)\cos (x)$

Possible Answers:

$(2x+1)-\sin (x)$

$(x^2+x)+\cos (x)$

$-(x^2+x)\sin (x)+(2x+1)\cos (x)$

$-(2x+1)\sin (x)$

Correct answer:

$-(x^2+x)\sin (x)+(2x+1)\cos (x)$

Explanation:

Apply the product rule:

$\frac{d}{dx}[f(x)*g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$

Then apply the product rule $nx^{n-1}$ (where "n" is the exponent) where needed.

$\\\frac{d}{dx}[(x^2+x)\cos (x)]\\ \\=(x^2+x)\frac{d}{dx}\cos (x)+\cos (x)\frac{d}{dx}(x^2+x)\\ \\=(x^2+x)(-\sin (x))+\cos (x)(2x^{2-1}+1x^{^{1-1}})\\ \\=-(x^2+x)\sin (x)+(2x+1)\cos (x)$

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Example Question #183 : How To Find Differential Functions

Differentiate the function:

$f(x)=(\cos (x)+\sin (x))(\sin (x)-\cos (x))$

Possible Answers:

$(\cos (x)+\sin (x))+(\sin (x)-\cos (x))$

$(\cos (x)+\sin (x))+(\cos (x)-\sin (x))$

$(\cos (x)+\sin (x))^2+(\sin (x)-\cos (x))(\cos (x)-\sin (x))$

$(\cos (x)+\sin (x))(\cos (x)-\sin (x))$

Correct answer:

$(\cos (x)+\sin (x))^2+(\sin (x)-\cos (x))(\cos (x)-\sin (x))$

Explanation:

Apply the product rule:

$\frac{d}{dx}[f(x)*g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$

$\\ \frac{d}{dx}[(\cos (x)+\sin (x))(\sin (x)-\cos (x))]\\ \\=(\cos (x)+\sin (x))\frac{d}{dx}(\sin (x)-\cos (x))+(\sin (x)-\cos (x))\frac{d}{dx}(\cos (x)+\sin (x))\\ \\=(\cos (x)+\sin (x))(\cos (x)-(-\sin (x)))+(\sin (x)-\cos (x))(-\sin (x)+\cos (x))\\ \\=(\cos (x)+\sin (x))^2+(\sin (x)-\cos (x))(\cos (x)-\sin (x))$

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Example Question #371 : Differential Functions

Differentiate the function:

f(x)=(x^3+3x)^2

Possible Answers:

2(x^2+3x)

2(x^3+3x)(3x+3)

(2x+3)

2(x^2+3x)+(2x+3)

Correct answer:

2(x^3+3x)(3x+3)

Explanation:

Tek the derivative of the function using the power rule $nx^{n-1}$ (where "n" is the exponent) on the original function then multiply that by the derivative with respect to "x" again using the power rule:

$\\ \frac{d}{dx}(x^3+3x)^2=2(x^3+3x)^{2-1}\frac{d}{dx}(x^3+3x)\\ \\=2(x^3+3x)(3x^{2-1}+3x^{1-1})\\ \\=2(x^3+3x)(3x+3)$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #172 : How To Find Differential Functions

Example Question #1391 : Calculus

Example Question #1391 : Calculus

Example Question #174 : How To Find Differential Functions

Example Question #175 : How To Find Differential Functions

Example Question #1392 : Calculus

Example Question #1391 : Calculus

Example Question #182 : How To Find Differential Functions

Example Question #183 : How To Find Differential Functions

Example Question #371 : Differential Functions

All Calculus 1 Resources

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