Partial Derivatives Practice
MathSci Problems Partial Derivatives Practice Questions ∂∂x(xeyz)= ?frac{partial}{partial x}big(xe^{yz}big)= ? ∂∂y(xeyz)= ?frac{partial}{partial y}big(xe^{yz}big)= ? ∂∂x ln(x2+y2)= ?frac{partial}{partial x},ln(x^2+y^2)= ? ∂∂y (x2y3+sin(xy))= ?frac{partial}{partial y},big(x^2y^3+sin(xy)big)= ? ∂2∂x ∂y exy= ?frac{partial^2}{partial x,partial y},e^{xy}= ? ∂∂x (x+yz)3= ?frac{partial}{partial x},(x+yz)^{3}= ? ∂∂x (y2cos(xy))= ?frac{partial}{partial x},big(y^{2}cos(xy)big)= ? ∂∂x xyzx+y+z= ?frac{partial}{partial x},frac{xyz}{x+y+z}= ? Evaluate at the point f(x,y)=x2+y2,∂f∂y(3,4)= ?f(x,y)=sqrt{x^{2}+y^{2}},qquad frac{partial f}{partial y}(3,4)= ? Solutions 1) Partial derivatives are not as hard as they look! Basically, just differentiate with […]
Derivatives Conceptual Practice
MathSci Problems Derivatives Conceptual Practice Questions ddx(e)= ?frac{d}{dx}(e)= ? ddx ∫01x dx= ?frac{d}{dx},int_{0}^{1} x,dx= ? cos ( ddx∫abf(x) dx)= ?cos!Big(,frac{d}{dx}int_{a}^{b} f(x),dxBig)= ? ddx ∫01x dt= ?frac{d}{dx},int_{0}^{1} x,dt= ? Solutions 1) ee is a constant, not exe^{x} . ddx(e)=0.frac{d}{dx}(e)=0. 2) A definite integral with fixed bounds is a constant. ∫01x dx=12⇒ddx(12)=0.int_{0}^{1} x,dx=tfrac12 quadRightarrowquad frac{d}{dx}(tfrac12)=0. 3) Inside the cosine, ddx∫abf(x) dx=0dfrac{d}{dx}int_{a}^{b} f(x),dx=0 . cos(0)=1.cos(0)=1. 4) Here the integral is with […]
Chain Rule Mixed Practice
MathSci Problems Chain Rule Mixed Practice Practice Problems Differentiate ddx ecos(3x)frac{d}{dx}, e^{cos(3x)} Differentiate ddx sin(x2+1)frac{d}{dx},sin(x^2+1) Differentiate ddx (cos(x3))2frac{d}{dx},big(cos(x^3)big)^2 Differentiate ddx ln(5×2+1)frac{d}{dx},ln(5x^2+1) Differentiate ddx 1+3×4 frac{d}{dx},sqrt{,1+3x^4,} Differentiate ddx (2x+1)10frac{d}{dx},(2x+1)^{10} Differentiate ddx arctan ( 1−x2 )(assume ∣x∣<1)frac{d}{dx},arctan!big(sqrt{,1-x^2,}big) qquad(text{assume }|x|<1) Differentiate ddx loga(3x+5)(a>0, a≠1)frac{d}{dx},log_a(3x+5) quad (a>0, aneq 1) Differentiate ddx e x2sinx frac{d}{dx},e^{,x^2sin x} Differentiate (away from points where sin(2x)=0sin(2x)=0) ddx ∣sin(2x)∣frac{d}{dx},big|sin(2x)big| If ffis differentiable and a,ba,bare constants, find ddx f(ax2+b)frac{d}{dx},f(ax^2+b) Solutions 1) ddx ecos(3x)=ecos(3x)⋅ddx(cos(3x))=ecos(3x)⋅(−sin(3x)⋅3)=−3sin(3x) ecos(3x).frac{d}{dx}, e^{cos(3x)} = […]