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The tangent is the straight line passing through x=1, touching the curve only at that point. For x=1, y=(4+1)^3=125 Using the chain rule we obtain dy/dx = 38x(4x^2+1)^2. To then get the gradient ...
I often find it difficult to remember all the different identities, so what is useful is instead to just remember the familiar identity sin^2(x) + cos^2(x) = 1 that we have come across many times, and div...
Recall the product rule for differentiation. If y=uv, where u and v are functions defined by functions of x, then we can take the derivative of y as: y'=u'v+v'u () (where ' denotes the derivative) ...
First, we use the trig identities: cos2x=cos^2x-sin^2x, cos^2x+sin^2x=1 and sin2x=2sinxcosx to transform the integral to ∫(cos2x)/(1+sin2x)dx.We know that ∫f'(x)/f(x)dx = ln|f(x)|+c, so we let f(x)=...
First, multiply throughout by the denominator of the main function to give as follows: 4-2x = A(x+1)(x+3) + B(2x+1)(x+3) + C(2x+1)(x+1) Then, choose values of x which will cause two of the constants to va...
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